FoxDifferential.Completed.Continuous.Universal.AugmentationQuotient
This module develops the augmentation side of the construction. It identifies the relevant kernels and ideals and compares the finite-stage and completed forms.
import
def zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard
(psi : ContinuousMonoidHom G H) :
Submodule (ZCCompletedGroupAlgebra C G)
(zcCompletedGroupAlgebraStandardAugmentationIdeal C G) :=
Submodule.span (ZCCompletedGroupAlgebra C G)
(Set.range fun p :
ProfiniteKernelSubgroup psi × zcCompletedGroupAlgebraStandardAugmentationIdeal C G =>
(zcGroupLike C G p.1.1 - 1) • p.2)The algebraic product \(I(\ker \psi)I(G)\) inside the algebraic standard source augmentation ideal.
theorem zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard_generator_mem
(psi : ContinuousMonoidHom G H) (n : ProfiniteKernelSubgroup psi)
(s : zcCompletedGroupAlgebraStandardAugmentationIdeal C G) :
(zcGroupLike C G n.1 - 1) • s ∈
zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psiShow proof
Submodule.subset_span (Set.mem_range_self (n, s))Proof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□def zcCompletedGroupAlgebraStandardAugmentationIdealProjection
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(i : ZCCompletedGroupAlgebraIndex C G) :
zcCompletedGroupAlgebraStandardAugmentationIdeal C G →
zcCompletedGroupAlgebraStageAugmentationIdeal C G i := by
intro x
let xAug : ZCCompletedGroupAlgebraAugmentationIdeal C G :=
⟨x, zcCompletedGroupAlgebraStandardAugmentationIdeal_le_augmentationIdeal C G x.2⟩
exact zcCompletedGroupAlgebraAugmentationIdealProjection C G i xAug
@[simp]Projection of the algebraic standard augmentation ideal to a finite augmentation stage.
theorem zcCompletedGroupAlgebraStandardAugmentationIdealProjection_val
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(i : ZCCompletedGroupAlgebraIndex C G)
(x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G) :
((zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i x :
ZCCompletedGroupAlgebraStage C G i)) =
zcCompletedGroupAlgebraProjection C G i xThe value of the augmentation-ideal projection is the value of the corresponding finite-stage projection.
Show proof
rflProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem continuous_zcCompletedGroupAlgebraStandardAugmentationIdealProjection
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(i : ZCCompletedGroupAlgebraIndex C G) :
Continuous (zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i)The standard augmentation-ideal projection is continuous.
Show proof
by
have hval : Continuous (fun x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G =>
zcCompletedGroupAlgebraProjection C G i (x : ZCCompletedGroupAlgebra C G)) :=
(continuous_zcCompletedGroupAlgebraProjection C G i).comp continuous_subtype_val
exact Continuous.subtype_mk hval
(fun x => (zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i x).2)Proof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem zcCompletedGAStageAugmentationIdeal_identityBoundary_monoidAlgebraToIdentity
(i : ZCCompletedGroupAlgebraIndex C G)
(x : zcCompletedGroupAlgebraStageAugmentationIdeal C G i) :
identityCrossedDifferentialBoundary
(monoidAlgebraToIdentityCrossedDifferentialModule
(S := ModNCompletedCoeff i.1.modulus)
(G := CompletedGroupAlgebraQuotientInClass G C i.2)
(x : ZCCompletedGroupAlgebraStage C G i)) =
(x : ZCCompletedGroupAlgebraStage C G i)On a finite coefficient stage, the identity crossed-differential boundary is a left inverse to the additive lift from the finite augmentation ideal.
Show proof
by
letI : Fact (0 < i.1.modulus) := ⟨i.1.positive⟩
have hxaug :
(MonoidAlgebra.lift
(ModNCompletedCoeff i.1.modulus)
(ModNCompletedCoeff i.1.modulus)
(CompletedGroupAlgebraQuotientInClass G C i.2)
(1 : CompletedGroupAlgebraQuotientInClass G C i.2 →* ModNCompletedCoeff i.1.modulus))
(x : ZCCompletedGroupAlgebraStage C G i) = 0 := by
simpa [modNCompletedGroupAlgebraStageAugmentationInClass] using
(mem_zcCompletedGroupAlgebraStageAugmentationIdeal_iff
(C := C) (H := G) (i := i) (x := (x : ZCCompletedGroupAlgebraStage C G i))).1 x.2
exact
idCrossedDiffBoundary_monoidAlgebraToModule_of_augmentation_eq_zero
(S := ModNCompletedCoeff i.1.modulus)
(G := CompletedGroupAlgebraQuotientInClass G C i.2)
(a := (x : ZCCompletedGroupAlgebraStage C G i))
hxaug
@[simp]Proof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem zcCompletedGroupAlgebraStandardAugmentationIdealProjection_zero
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(i : ZCCompletedGroupAlgebraIndex C G) :
zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i
(0 : zcCompletedGroupAlgebraStandardAugmentationIdeal C G) = 0The finite-stage augmentation-ideal projection is compatible with zero.
Show proof
by
apply Subtype.ext
simp only [zcCompletedGroupAlgebraStandardAugmentationIdealProjection_val, ZeroMemClass.coe_zero,
zcCompletedGroupAlgebraProjection_zero]
@[simp]Proof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem zcCompletedGroupAlgebraStandardAugmentationIdealProjection_add
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(i : ZCCompletedGroupAlgebraIndex C G)
(x y : zcCompletedGroupAlgebraStandardAugmentationIdeal C G) :
zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i (x + y) =
zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i x +
zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i yThe finite-stage augmentation-ideal projection is compatible with addition.
Show proof
by
apply Subtype.ext
simp only [zcCompletedGroupAlgebraStandardAugmentationIdealProjection_val, Submodule.coe_add,
zcCompletedGroupAlgebraProjection_add]
@[simp]Proof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem zcCompletedGroupAlgebraStandardAugmentationIdealProjection_smul
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(i : ZCCompletedGroupAlgebraIndex C G)
(a : ZCCompletedGroupAlgebra C G)
(x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G) :
zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i (a • x) =
zcCompletedGroupAlgebraProjection C G i a •
zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i xThe finite-stage augmentation-ideal projection is compatible with scalar multiplication.
Show proof
by
apply Subtype.ext
change zcCompletedGroupAlgebraProjection C G i (a * (x : ZCCompletedGroupAlgebra C G)) =
zcCompletedGroupAlgebraProjection C G i a *
zcCompletedGroupAlgebraProjection C G i (x : ZCCompletedGroupAlgebra C G)
rw [zcCompletedGroupAlgebraProjection_mul]Proof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□def zcCompletedGroupAlgebraStandardAugmentationIdealProjectionLinear
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(i : ZCCompletedGroupAlgebraIndex C G) :
zcCompletedGroupAlgebraStandardAugmentationIdeal C G →ₛₗ[
zcCompletedGroupAlgebraProjectionRingHom C G i]
zcCompletedGroupAlgebraStageAugmentationIdeal C G i where
toFun := zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i
map_add' := by
intro x y
exact zcCompletedGroupAlgebraStandardAugmentationIdealProjection_add C i x y
map_smul' := by
intro a x
exact zcCompletedGroupAlgebraStandardAugmentationIdealProjection_smul C i a x
@[simp]The finite-stage projection of the standard augmentation ideal, as a semilinear map over the completed group-algebra projection.
theorem zcCompletedGroupAlgebraStandardAugmentationIdealProjectionLinear_apply
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(i : ZCCompletedGroupAlgebraIndex C G)
(x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G) :
zcCompletedGroupAlgebraStandardAugmentationIdealProjectionLinear C i x =
zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i xThe augmentation map is evaluated by projecting to the corresponding finite group-algebra stage and summing coefficients via the usual group-algebra augmentation.
Show proof
rfl
@[simp]Proof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem zcCompletedGroupAlgebraStandardAugmentationIdealProjection_sub
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(i : ZCCompletedGroupAlgebraIndex C G)
(x y : zcCompletedGroupAlgebraStandardAugmentationIdeal C G) :
zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i (x - y) =
zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i x -
zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i yThe finite-stage augmentation-ideal projection is compatible with subtraction.
Show proof
by
simpa using
map_sub (zcCompletedGroupAlgebraStandardAugmentationIdealProjectionLinear C i) x yProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□def zcCompletedGroupAlgebraStandardAugmentationIdealProjectionProduct
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C] :
zcCompletedGroupAlgebraStandardAugmentationIdeal C G →
∀ i : ZCCompletedGroupAlgebraIndex C G,
zcCompletedGroupAlgebraStageAugmentationIdeal C G i :=
fun x i => zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i x
@[simp]The product of all finite-stage projections of the standard completed augmentation ideal.
theorem zcCompletedGroupAlgebraStandardAugmentationIdealProjectionProduct_apply
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G)
(i : ZCCompletedGroupAlgebraIndex C G) :
zcCompletedGroupAlgebraStandardAugmentationIdealProjectionProduct C x i =
zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i xThe augmentation map is evaluated by projecting to the corresponding finite group-algebra stage and summing coefficients via the usual group-algebra augmentation.
Show proof
rflProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem zcCompletedGroupAlgebraStandardAugmentationIdealProjectionProduct_injective
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C] :
Function.Injective
(zcCompletedGroupAlgebraStandardAugmentationIdealProjectionProduct C (G := G))Finite-stage projections separate points of the standard completed augmentation ideal.
Show proof
by
intro x y hxy
apply Subtype.ext
apply Subtype.ext
funext i
exact congrArg Subtype.val (congrFun hxy i)Proof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem zcCompletedGroupAlgebraStandardAugmentationIdealProjection_ext
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
{x y : zcCompletedGroupAlgebraStandardAugmentationIdeal C G}
(h : ∀ i : ZCCompletedGroupAlgebraIndex C G,
zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i x =
zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i y) :
x = yExtensionality for standard completed augmentation ideal elements by finite-stage projections.
Show proof
zcCompletedGroupAlgebraStandardAugmentationIdealProjectionProduct_injective C
(by
funext i
exact h i)Proof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem zcCompletedGroupAlgebraStandardAugmentationIdealProjection_surjective
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(i : ZCCompletedGroupAlgebraIndex C G) :
Function.Surjective
(zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i)Every finite standard augmentation stage is hit by the completed standard augmentation ideal projection.
Show proof
by
intro x
rcases zcCompletedGroupAlgebraStageAugmentationIdeal_mem_projection_standard
(C := C) (H := G) i x with
⟨y, hy, hproj⟩
refine ⟨⟨y, hy⟩, ?_⟩
apply Subtype.ext
simpa [zcCompletedGroupAlgebraStandardAugmentationIdealProjection_val] using hprojProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□def zcCompletedGroupAlgebraOpenImageKernelAugmentationIdealMulStageStandard
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(i : ZCCompletedGroupAlgebraIndex C G) :
Submodule (ZCCompletedGroupAlgebraStage C G i)
(zcCompletedGroupAlgebraStageAugmentationIdeal C G i) :=
Submodule.span (ZCCompletedGroupAlgebraStage C G i)
(Set.range fun p :
(zcCompletedGroupAlgebraOpenImageQuotientMap C hC hForm psi hpsi hfopen i).ker ×
zcCompletedGroupAlgebraStageAugmentationIdeal C G i =>
(MonoidAlgebra.of (ModNCompletedCoeff i.1.modulus)
(CompletedGroupAlgebraQuotientInClass G C i.2) p.1.1 - 1) • p.2)theorem zcCompletedGAOpenImageKernelAugmentationIdealMulStageStandard_generator_mem
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(i : ZCCompletedGroupAlgebraIndex C G)
(q : (zcCompletedGroupAlgebraOpenImageQuotientMap C hC hForm psi hpsi hfopen i).ker)
(s : zcCompletedGroupAlgebraStageAugmentationIdeal C G i) :
(MonoidAlgebra.of (ModNCompletedCoeff i.1.modulus)
(CompletedGroupAlgebraQuotientInClass G C i.2) q.1 - 1) • s ∈
zcCompletedGroupAlgebraOpenImageKernelAugmentationIdealMulStageStandard
C hC hForm psi hpsi hfopen iShow proof
Submodule.subset_span (Set.mem_range_self (q, s))Proof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□def zcCompletedGroupAlgebraOpenImageKernelClass
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(i : ZCCompletedGroupAlgebraIndex C G)
(n : ProfiniteKernelSubgroup psi) :
(zcCompletedGroupAlgebraOpenImageQuotientMap C hC hForm psi hpsi hfopen i).ker := by
refine ⟨QuotientGroup.mk'
((((OrderDual.ofDual i.2).1 : OpenNormalSubgroup G) : Subgroup G)) n.1, ?_⟩
rw [MonoidHom.mem_ker]
rw [zcCompletedGroupAlgebraOpenImageQuotientMap_mk]
change QuotientGroup.mk'
((((OrderDual.ofDual
(zcCompletedGroupAlgebraOpenImageIndexInClass C hForm psi hpsi hfopen i)).1 :
OpenNormalSubgroup H) : Subgroup H)) (psi n.1) = 1
rw [show psi n.1 = 1 from n.2]
simp only [QuotientGroup.mk'_apply, QuotientGroup.mk_one]
@[simp]An actual kernel element determines a class in the finite-stage open-image kernel.
theorem zcCompletedGroupAlgebraOpenImageKernelClass_val
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(i : ZCCompletedGroupAlgebraIndex C G)
(n : ProfiniteKernelSubgroup psi) :
(zcCompletedGroupAlgebraOpenImageKernelClass C hC hForm psi hpsi hfopen i n).1 =
QuotientGroup.mk'
((((OrderDual.ofDual i.2).1 : OpenNormalSubgroup G) : Subgroup G)) n.1The open-image kernel class has the expected underlying subgroup, characterized by finite-stage Fox coordinate formulas.
Show proof
rfl
@[simp 900]Proof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem zcCompletedGroupAlgebraStandardAugmentationIdealProjection_kernel_generator_smul
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(i : ZCCompletedGroupAlgebraIndex C G)
(n : ProfiniteKernelSubgroup psi)
(s : zcCompletedGroupAlgebraStandardAugmentationIdeal C G) :
zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i
((zcGroupLike C G n.1 - 1) • s) =
(MonoidAlgebra.of (ModNCompletedCoeff i.1.modulus)
(CompletedGroupAlgebraQuotientInClass G C i.2)
(zcCompletedGroupAlgebraOpenImageKernelClass
C hC hForm psi hpsi hfopen i n).1 - 1) •
zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i sThe finite-stage augmentation-ideal projection is compatible with scalar multiplication.
Show proof
by
apply Subtype.ext
change zcCompletedGroupAlgebraProjection C G i
((zcGroupLike C G n.1 - 1) * (s : ZCCompletedGroupAlgebra C G)) =
(MonoidAlgebra.of (ModNCompletedCoeff i.1.modulus)
(CompletedGroupAlgebraQuotientInClass G C i.2)
(QuotientGroup.mk'
((((OrderDual.ofDual i.2).1 : OpenNormalSubgroup G) : Subgroup G)) n.1) - 1) *
zcCompletedGroupAlgebraProjection C G i (s : ZCCompletedGroupAlgebra C G)
rw [zcCompletedGroupAlgebraProjection_mul]
simp only [zcCompletedGroupAlgebraProjection_sub, zcCompletedGroupAlgebraProjection_groupLike,
MonoidAlgebra.of_apply, zcCompletedGroupAlgebraProjection_one, QuotientGroup.mk'_apply]Proof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem zcCompletedGAKernelAugmentationIdealMulStandard_proj_mem_openImageStage
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(i : ZCCompletedGroupAlgebraIndex C G)
{x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G}
(hx : x ∈ zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi) :
zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i x ∈
zcCompletedGroupAlgebraOpenImageKernelAugmentationIdealMulStageStandard
C hC hForm psi hpsi hfopen iProjecting an element of \(I(\ker\psi)I(G)\) to an open-image finite stage lands in the corresponding finite-stage kernel-augmentation product.
Show proof
by
let T :=
zcCompletedGroupAlgebraOpenImageKernelAugmentationIdealMulStageStandard
C hC hForm psi hpsi hfopen i
refine Submodule.span_induction
(p := fun x _ => zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i x ∈ T)
?_ ?_ ?_ ?_ hx
· rintro _ ⟨p, rfl⟩
rcases p with ⟨n, s⟩
rw [zcCompletedGroupAlgebraStandardAugmentationIdealProjection_kernel_generator_smul
C hC hForm psi hpsi hfopen i n s]
exact
zcCompletedGAOpenImageKernelAugmentationIdealMulStageStandard_generator_mem
C hC hForm psi hpsi hfopen i
(zcCompletedGroupAlgebraOpenImageKernelClass C hC hForm psi hpsi hfopen i n)
(zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i s)
· simp only [zcCompletedGroupAlgebraStandardAugmentationIdealProjection_zero, zero_mem, T]
· intro x y _ _ hx hy
simpa [T] using T.add_mem hx hy
· intro a x _ hx
rw [zcCompletedGroupAlgebraStandardAugmentationIdealProjection_smul]
exact T.smul_mem (zcCompletedGroupAlgebraProjection C G i a) hxtheorem zcCompletedGAOpenImageKernelAugmentationIdealMulStageStandard_mem_proj
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(i : ZCCompletedGroupAlgebraIndex C G)
(x : zcCompletedGroupAlgebraStageAugmentationIdeal C G i)
(hx :
x ∈ zcCompletedGroupAlgebraOpenImageKernelAugmentationIdealMulStageStandard
C hC hForm psi hpsi hfopen i) :
∃ y ∈ zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi,
zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i y = xEvery element of the finite-stage open-image kernel-augmentation product is the projection of an element of the algebraic product \(I(\ker\psi)I(G)\).
Show proof
by
let T :=
zcCompletedGroupAlgebraOpenImageKernelAugmentationIdealMulStageStandard
C hC hForm psi hpsi hfopen i
let P : zcCompletedGroupAlgebraStageAugmentationIdeal C G i → Prop := fun x =>
∃ y ∈ zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi,
zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i y = x
refine Submodule.span_induction (p := fun x _ => P x) ?_ ?_ ?_ ?_ hx
· rintro _ ⟨p, rfl⟩
rcases p with ⟨q, s⟩
rcases
zcCompletedGroupAlgebraOpenImageQuotientMap_kernel_lift
C hC hForm psi hpsi hfopen i q with
⟨n, hn⟩
rcases
zcCompletedGroupAlgebraStageAugmentationIdeal_mem_projection_standard
(C := C) (H := G) i s with
⟨s', hs', hs'proj⟩
let sStd : zcCompletedGroupAlgebraStandardAugmentationIdeal C G := ⟨s', hs'⟩
refine ⟨(zcGroupLike C G n.1 - 1) • sStd,
zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard_generator_mem C psi n sStd, ?_⟩
rw [zcCompletedGroupAlgebraStandardAugmentationIdealProjection_kernel_generator_smul
C hC hForm psi hpsi hfopen i n sStd]
apply Subtype.ext
change
(MonoidAlgebra.of (ModNCompletedCoeff i.1.modulus)
(CompletedGroupAlgebraQuotientInClass G C i.2)
(QuotientGroup.mk'
((((OrderDual.ofDual i.2).1 : OpenNormalSubgroup G) : Subgroup G)) n.1) - 1) *
zcCompletedGroupAlgebraProjection C G i s' =
(MonoidAlgebra.of (ModNCompletedCoeff i.1.modulus)
(CompletedGroupAlgebraQuotientInClass G C i.2) q.1 - 1) * s.1
rw [hn, hs'proj]
· refine ⟨0, (zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi).zero_mem, ?_⟩
simp only [zcCompletedGroupAlgebraStandardAugmentationIdealProjection_zero]
· intro x y _ _ hx hy
rcases hx with ⟨x', hx'mem, hx'proj⟩
rcases hy with ⟨y', hy'mem, hy'proj⟩
refine ⟨x' + y',
(zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi).add_mem hx'mem hy'mem, ?_⟩
simp only [zcCompletedGroupAlgebraStandardAugmentationIdealProjection_add, hx'proj, hy'proj]
· intro a x _ hx
rcases hx with ⟨x', hx'mem, hx'proj⟩
rcases zcCompletedGroupAlgebraProjection_surjective C G i a with ⟨a', ha'⟩
refine ⟨a' • x',
(zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi).smul_mem a' hx'mem, ?_⟩
rw [zcCompletedGroupAlgebraStandardAugmentationIdealProjection_smul, ha', hx'proj]Proof. Induct over the finite-stage span. For a generator, lift the finite-stage kernel element through the open surjective map and lift the augmentation-ideal factor to the standard completed augmentation ideal; the projection formula gives the required equality. The submodule operations handle zero, addition, and scalar multiples.
□def zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi) :
Submodule (ZCCompletedGroupAlgebra C G)
(zcCompletedGroupAlgebraStandardAugmentationIdeal C G) where
carrier := {x | ∀ i : ZCCompletedGroupAlgebraIndex C G,
zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i x ∈
zcCompletedGroupAlgebraOpenImageKernelAugmentationIdealMulStageStandard
C hC hForm psi hpsi hfopen i}
zero_mem' := by
intro i
simp only [zcCompletedGroupAlgebraStandardAugmentationIdealProjection_zero, zero_mem]
add_mem' := by
intro x y hx hy i
simpa using
(zcCompletedGroupAlgebraOpenImageKernelAugmentationIdealMulStageStandard
C hC hForm psi hpsi hfopen i).add_mem (hx i) (hy i)
smul_mem' := by
intro a x hx i
rw [zcCompletedGroupAlgebraStandardAugmentationIdealProjection_smul]
exact
(zcCompletedGroupAlgebraOpenImageKernelAugmentationIdealMulStageStandard
C hC hForm psi hpsi hfopen i).smul_mem
(zcCompletedGroupAlgebraProjection C G i a) (hx i)The closed finite-stage hull of \(I(\ker \psi)I(G)\) inside the standard source augmentation ideal: an element belongs exactly when every finite projection belongs to the finite-stage open-image kernel product.
theorem zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard_le_closed
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi) :
zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi ≤
zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopenThe algebraic product is contained in its finite-stage closed hull.
Show proof
by
intro x hx i
exact
zcCompletedGAKernelAugmentationIdealMulStandard_proj_mem_openImageStage
C hC hForm psi hpsi hfopen i hxProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem isClosed_zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi) :
IsClosed
((zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen :
Set (zcCompletedGroupAlgebraStandardAugmentationIdeal C G)))The finite-stage hull is closed in the standard source augmentation ideal.
Show proof
by
change IsClosed {x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G |
∀ i : ZCCompletedGroupAlgebraIndex C G,
zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i x ∈
zcCompletedGroupAlgebraOpenImageKernelAugmentationIdealMulStageStandard
C hC hForm psi hpsi hfopen i}
simp only [Set.setOf_forall]
refine isClosed_iInter ?_
intro i
haveI : DiscreteTopology (zcCompletedGroupAlgebraStageAugmentationIdeal C G i) := by
infer_instance
exact
(isClosed_discrete
((zcCompletedGroupAlgebraOpenImageKernelAugmentationIdealMulStageStandard
C hC hForm psi hpsi hfopen i :
Set (zcCompletedGroupAlgebraStageAugmentationIdeal C G i)))).preimage
(continuous_zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i)Proof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem closure_zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard_le_closed
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi) :
closure
((zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi :
Set (zcCompletedGroupAlgebraStandardAugmentationIdeal C G))) ⊆
(zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen :
Set (zcCompletedGroupAlgebraStandardAugmentationIdeal C G))The closure of the algebraic product is contained in the finite-stage closed hull.
Show proof
by
exact
closure_minimal
(by
intro x hx
exact
zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard_le_closed
C hC hForm psi hpsi hfopen hx)
(isClosed_zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen)Proof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed_le_closure
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi) :
(zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen :
Set (zcCompletedGroupAlgebraStandardAugmentationIdeal C G)) ⊆
closure
((zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi :
Set (zcCompletedGroupAlgebraStandardAugmentationIdeal C G)))The finite-stage closed hull is contained in the closure of the algebraic product.
Show proof
by
intro x hx
let R := ZCCompletedGroupAlgebra C G
let Ssys := zcCompletedGroupAlgebraSystem C G
let Ystd : Set (zcCompletedGroupAlgebraStandardAugmentationIdeal C G) :=
zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi
let Yamb : Set R := Subtype.val '' Ystd
have hxAmb : (x : R) ∈ closure Yamb := by
letI : Nonempty (ZCCompletedGroupAlgebraIndex C G) :=
⟨(ProCGroups.Completion.ProCIntegerIndex.terminal (C := C) inferInstance,
zcCompletedGroupAlgebraTopIndex C G)⟩
letI : ∀ i : ZCCompletedGroupAlgebraIndex C G, TopologicalSpace (Ssys.X i) := fun _ =>
inferInstance
letI : ∀ i : ZCCompletedGroupAlgebraIndex C G, CompactSpace (Ssys.X i) := fun i => by
dsimp [Ssys, zcCompletedGroupAlgebraSystem]
infer_instance
letI : ∀ i : ZCCompletedGroupAlgebraIndex C G, T2Space (Ssys.X i) := fun i => by
dsimp [Ssys, zcCompletedGroupAlgebraSystem]
infer_instance
have hdir : Directed (· ≤ ·) (id : ZCCompletedGroupAlgebraIndex C G →
ZCCompletedGroupAlgebraIndex C G) :=
directed_zcCompletedGroupAlgebraIndex (C := C) (H := G) hForm
rw [Ssys.mem_isClosed_iff_forall_projection_mem hdir isClosed_closure]
intro i
rcases
zcCompletedGAOpenImageKernelAugmentationIdealMulStageStandard_mem_proj
C hC hForm psi hpsi hfopen i
(zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i x)
(hx i) with
⟨y, hy, hyproj⟩
refine ⟨(y : R), subset_closure ?_, ?_⟩
· exact ⟨y, by simpa [Ystd] using hy, rfl⟩
· simpa [Ssys, zcCompletedGroupAlgebraSystem,
zcCompletedGroupAlgebraStandardAugmentationIdealProjection_val] using
congrArg Subtype.val hyproj
have hclosure :
closure Ystd =
(Subtype.val : zcCompletedGroupAlgebraStandardAugmentationIdeal C G → R) ⁻¹'
closure Yamb := by
exact Topology.IsEmbedding.subtypeVal.closure_eq_preimage_closure_image Ystd
show x ∈ closure Ystd
rw [hclosure]
exact hxAmbProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem closure_zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard_eq_closed
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi) :
closure
((zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi :
Set (zcCompletedGroupAlgebraStandardAugmentationIdeal C G))) =
(zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen :
Set (zcCompletedGroupAlgebraStandardAugmentationIdeal C G))Show proof
by
exact Set.Subset.antisymm
(closure_zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard_le_closed
C hC hForm psi hpsi hfopen)
(zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed_le_closure
C hC hForm psi hpsi hfopen)Proof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem isClosed_zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard_iff_eq_closed
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi) :
IsClosed
((zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi :
Set (zcCompletedGroupAlgebraStandardAugmentationIdeal C G))) ↔
(zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi :
Set (zcCompletedGroupAlgebraStandardAugmentationIdeal C G)) =
(zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen :
Set (zcCompletedGroupAlgebraStandardAugmentationIdeal C G))The algebraic product is closed exactly when it already equals the finite-stage closed hull. This is the precise non-circular closedness frontier.
Show proof
by
constructor
· intro hclosed
apply Set.Subset.antisymm
· exact zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard_le_closed
C hC hForm psi hpsi hfopen
· intro x hx
have hxclosure :
x ∈ closure
((zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi :
Set (zcCompletedGroupAlgebraStandardAugmentationIdeal C G))) := by
rw [closure_zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard_eq_closed
C hC hForm psi hpsi hfopen]
exact hx
rwa [hclosed.closure_eq] at hxclosure
· intro hEq
rw [hEq]
exact isClosed_zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopenProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□abbrev KernelAugmentationIdealQuotient
(psi : ContinuousMonoidHom G H) : Type u :=
zcCompletedGroupAlgebraStandardAugmentationIdeal C G ⧸
zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psiThe source-side augmentation quotient \(I\mathbb{Z}_C\llbracket G\rrbracket / I(\ker \psi) I\mathbb{Z}_C\llbracket G\rrbracket\), before descending scalars to \(\mathbb{Z}_C\llbracket H\rrbracket\).
abbrev KernelAugmentationIdealClosedQuotient
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi) : Type u :=
zcCompletedGroupAlgebraStandardAugmentationIdeal C G ⧸
zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopenThe closed source-side augmentation quotient, using the finite-stage closed hull of \(I(\ker \psi)I(G)\) as denominator.
theorem isQuotientMap_kernelAugmentationIdealClosedQuotient_mkQ
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi) :
Topology.IsQuotientMap
(zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen).mkQThe quotient map to the closed source augmentation quotient is a quotient map.
Show proof
by
rw [Topology.isQuotientMap_iff]
constructor
· exact
Submodule.Quotient.mk_surjective
(p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen)
· intro s
rflProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem continuous_kernelAugmentationIdealClosedQuotient_iff_comp_mkQ
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
{A : Type u} [TopologicalSpace A]
(f : KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen → A) :
Continuous f ↔
Continuous (fun x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G =>
f ((zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen).mkQ x))Continuity out of the closed source augmentation quotient can be tested after precomposing with the quotient map from the standard augmentation ideal.
Show proof
by
simpa [Function.comp_def] using
(isQuotientMap_kernelAugmentationIdealClosedQuotient_mkQ
C hC hForm psi hpsi hfopen).continuous_iff (g := f)Proof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem t1Space_kernelAugmentationIdealClosedQuotient
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi) :
T1Space (KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)The closed source augmentation quotient is \(T_1\) for the quotient topology.
Show proof
by
letI : IsClosed
((zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen :
Submodule (ZCCompletedGroupAlgebra C G)
(zcCompletedGroupAlgebraStandardAugmentationIdeal C G)) :
Set (zcCompletedGroupAlgebraStandardAugmentationIdeal C G)) :=
isClosed_zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen
infer_instanceProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem isClosed_zero_kernelAugmentationIdealClosedQuotient
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi) :
IsClosed
({0} : Set
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen))The zero class is closed in the closed source augmentation quotient.
Show proof
by
letI : T1Space
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
t1Space_kernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen
exact isClosed_singletonProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□abbrev KernelAugmentationIdealClosedStageQuotient
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(i : ZCCompletedGroupAlgebraIndex C G) : Type u :=
zcCompletedGroupAlgebraStageAugmentationIdeal C G i ⧸
zcCompletedGroupAlgebraOpenImageKernelAugmentationIdealMulStageStandard
C hC hForm psi hpsi hfopen iThe finite-stage quotient of the source augmentation ideal by the open-image kernel product.
def kernelAugmentationIdealClosedQuotientStageProjection
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(i : ZCCompletedGroupAlgebraIndex C G) :
KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen →ₛₗ[
zcCompletedGroupAlgebraProjectionRingHom C G i]
KernelAugmentationIdealClosedStageQuotient C hC hForm psi hpsi hfopen i :=
(zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen).mapQ
(zcCompletedGroupAlgebraOpenImageKernelAugmentationIdealMulStageStandard
C hC hForm psi hpsi hfopen i)
(zcCompletedGroupAlgebraStandardAugmentationIdealProjectionLinear C i)
(by
intro x hx
exact hx i)
@[simp 900]The finite-stage coordinate of the closed source augmentation quotient.
theorem kernelAugmentationIdealClosedQuotientStageProjection_mk
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(i : ZCCompletedGroupAlgebraIndex C G)
(x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G) :
kernelAugmentationIdealClosedQuotientStageProjection
C hC hForm psi hpsi hfopen i
((zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen).mkQ x) =
((zcCompletedGroupAlgebraOpenImageKernelAugmentationIdealMulStageStandard
C hC hForm psi hpsi hfopen i).mkQ
(zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i x) :
KernelAugmentationIdealClosedStageQuotient C hC hForm psi hpsi hfopen i)The stage projection of a closed kernel-augmentation quotient class is represented by the corresponding finite-stage projection.
Show proof
by
exact
Submodule.mapQ_apply
(zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen)
(zcCompletedGroupAlgebraOpenImageKernelAugmentationIdealMulStageStandard
C hC hForm psi hpsi hfopen i)
(zcCompletedGroupAlgebraStandardAugmentationIdealProjectionLinear C i)
xProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem continuous_kernelAugmentationIdealClosedQuotientStageProjection
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(i : ZCCompletedGroupAlgebraIndex C G) :
Continuous
(kernelAugmentationIdealClosedQuotientStageProjection
C hC hForm psi hpsi hfopen i)Each finite-stage coordinate of the closed source augmentation quotient is continuous.
Show proof
by
rw [continuous_kernelAugmentationIdealClosedQuotient_iff_comp_mkQ
(C := C) (G := G) (H := H) hC hForm psi hpsi hfopen]
have hproj :
Continuous (zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i) :=
continuous_zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i
have hq :
Continuous (fun y : zcCompletedGroupAlgebraStageAugmentationIdeal C G i =>
((zcCompletedGroupAlgebraOpenImageKernelAugmentationIdealMulStageStandard
C hC hForm psi hpsi hfopen i).mkQ y :
KernelAugmentationIdealClosedStageQuotient C hC hForm psi hpsi hfopen i)) :=
continuous_quotient_mk'
simpa using hq.comp hprojProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□def kernelAugmentationIdealClosedQuotientStageProjectionProduct
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi) :
KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen →
∀ i : ZCCompletedGroupAlgebraIndex C G,
KernelAugmentationIdealClosedStageQuotient C hC hForm psi hpsi hfopen i :=
fun x i => kernelAugmentationIdealClosedQuotientStageProjection
C hC hForm psi hpsi hfopen i x
@[simp]The product of all finite-stage coordinates of the closed source augmentation quotient.
theorem kernelAugmentationIdealClosedQuotientStageProjectionProduct_apply
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(x : KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)
(i : ZCCompletedGroupAlgebraIndex C G) :
kernelAugmentationIdealClosedQuotientStageProjectionProduct
C hC hForm psi hpsi hfopen x i =
kernelAugmentationIdealClosedQuotientStageProjection
C hC hForm psi hpsi hfopen i xThe augmentation map is evaluated by projecting to the corresponding finite group-algebra stage and summing coefficients via the usual group-algebra augmentation.
Show proof
rflProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem continuous_kernelAugmentationIdealClosedQuotientStageProjectionProduct
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi) :
Continuous
(kernelAugmentationIdealClosedQuotientStageProjectionProduct
C hC hForm psi hpsi hfopen)The finite-stage coordinate product of the closed source augmentation quotient is continuous.
Show proof
by
exact continuous_pi fun i =>
continuous_kernelAugmentationIdealClosedQuotientStageProjection
C hC hForm psi hpsi hfopen iProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem kernelAugmentationIdealClosedQuotientStageProjectionProduct_injective
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi) :
Function.Injective
(kernelAugmentationIdealClosedQuotientStageProjectionProduct
C hC hForm psi hpsi hfopen)Finite-stage coordinates separate points in the closed source augmentation quotient.
Show proof
by
let S :=
zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen
intro qx qy hxy
revert qy
refine Submodule.Quotient.induction_on (p := S) qx ?_
intro x qy hxy
revert hxy
refine Submodule.Quotient.induction_on (p := S) qy ?_
intro y hxy
apply (Submodule.Quotient.eq S).2
change x - y ∈ S
intro i
let T :=
zcCompletedGroupAlgebraOpenImageKernelAugmentationIdealMulStageStandard
C hC hForm psi hpsi hfopen i
have hi :
kernelAugmentationIdealClosedQuotientStageProjection
C hC hForm psi hpsi hfopen i (Submodule.Quotient.mk (p := S) x) =
kernelAugmentationIdealClosedQuotientStageProjection
C hC hForm psi hpsi hfopen i (Submodule.Quotient.mk (p := S) y) := by
exact congrFun hxy i
have hstage :
zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i x -
zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i y ∈ T := by
apply (Submodule.Quotient.eq T).1
simpa [S, T] using hi
simpa [S, T] using hstageProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem kernelAugmentationIdealClosedQuotientStageProjection_ext
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
{x y : KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen}
(h : ∀ i : ZCCompletedGroupAlgebraIndex C G,
kernelAugmentationIdealClosedQuotientStageProjection
C hC hForm psi hpsi hfopen i x =
kernelAugmentationIdealClosedQuotientStageProjection
C hC hForm psi hpsi hfopen i y) :
x = yExtensionality for the closed source augmentation quotient by finite-stage coordinates.
Show proof
by
exact
kernelAugmentationIdealClosedQuotientStageProjectionProduct_injective
C hC hForm psi hpsi hfopen
(by
funext i
exact h i)Proof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem kernelAugmentationIdealClosedQuotient_topology_eq_induced_stageProjProduct
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi) :
(inferInstance :
TopologicalSpace
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)) =
TopologicalSpace.induced
(kernelAugmentationIdealClosedQuotientStageProjectionProduct
C hC hForm psi hpsi hfopen) inferInstanceThe quotient topology on the closed source augmentation quotient is exactly the topology induced by all finite closed-quotient coordinates.
Show proof
by
let Sclosed :=
zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen
let Q := KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen
let stageProduct :=
kernelAugmentationIdealClosedQuotientStageProjectionProduct
C hC hForm psi hpsi hfopen
ext U
constructor
· intro hU
let Tind : TopologicalSpace Q :=
TopologicalSpace.induced stageProduct inferInstance
rw [@isOpen_iff_forall_mem_open Q Tind U]
intro qx hqxU
refine Submodule.Quotient.induction_on
(p := Sclosed)
(C := fun qx =>
qx ∈ U → ∃ t, t ⊆ U ∧ @IsOpen Q Tind t ∧ qx ∈ t)
qx ?_ hqxU
intro x hxU
let q : zcCompletedGroupAlgebraStandardAugmentationIdeal C G → Q :=
Sclosed.mkQ
have hquot :
@Topology.IsQuotientMap
(zcCompletedGroupAlgebraStandardAugmentationIdeal C G) Q
inferInstance (QuotientModule.Quotient.topologicalSpace Sclosed) q := by
simpa [q, Sclosed] using
isQuotientMap_kernelAugmentationIdealClosedQuotient_mkQ
C hC hForm psi hpsi hfopen
have hUquot :
@IsOpen Q (QuotientModule.Quotient.topologicalSpace Sclosed) U := hU
have hpreOpen :
@IsOpen
(zcCompletedGroupAlgebraStandardAugmentationIdeal C G)
inferInstance (q ⁻¹' U) := by
letI : TopologicalSpace Q := QuotientModule.Quotient.topologicalSpace Sclosed
exact ((Topology.isQuotientMap_iff.mp hquot).2 U).1 hUquot
rcases isOpen_induced_iff.mp hpreOpen with ⟨V, hVopen, hVeq⟩
have hxV : (x : ZCCompletedGroupAlgebra C G) ∈ V := by
have hxpre : x ∈ q ⁻¹' U := hxU
rwa [← hVeq] at hxpre
let Ssys := zcCompletedGroupAlgebraSystem C G
letI : Nonempty (ZCCompletedGroupAlgebraIndex C G) :=
⟨(ProCGroups.Completion.ProCIntegerIndex.terminal (C := C) inferInstance,
zcCompletedGroupAlgebraTopIndex C G)⟩
letI : ∀ i : ZCCompletedGroupAlgebraIndex C G, TopologicalSpace (Ssys.X i) := fun _ =>
inferInstance
letI : ∀ i : ZCCompletedGroupAlgebraIndex C G, CompactSpace (Ssys.X i) := fun i => by
dsimp [Ssys, zcCompletedGroupAlgebraSystem]
infer_instance
letI : ∀ i : ZCCompletedGroupAlgebraIndex C G, T2Space (Ssys.X i) := fun i => by
dsimp [Ssys, zcCompletedGroupAlgebraSystem]
infer_instance
have hdir : Directed (· ≤ ·) (id : ZCCompletedGroupAlgebraIndex C G →
ZCCompletedGroupAlgebraIndex C G) :=
directed_zcCompletedGroupAlgebraIndex (C := C) (H := G) hForm
rcases Ssys.exists_projection_preimage_subset hdir hVopen hxV with
⟨i, W, hWopen, hxW, hWV⟩
let t : Set Q :=
{z | kernelAugmentationIdealClosedQuotientStageProjection
C hC hForm psi hpsi hfopen i z =
kernelAugmentationIdealClosedQuotientStageProjection
C hC hForm psi hpsi hfopen i (q x)}
refine ⟨t, ?_, ?_, ?_⟩
· intro z hz
refine Submodule.Quotient.induction_on
(p := Sclosed)
(C := fun z => z ∈ t → z ∈ U)
z ?_ hz
intro y hy
let T :=
zcCompletedGroupAlgebraOpenImageKernelAugmentationIdealMulStageStandard
C hC hForm psi hpsi hfopen i
have hyStage :
zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i y -
zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i x ∈ T := by
apply (Submodule.Quotient.eq T).1
have hy' := hy
dsimp [t, q] at hy'
simpa [Sclosed, T] using hy'
rcases
zcCompletedGAOpenImageKernelAugmentationIdealMulStageStandard_mem_proj
C hC hForm psi hpsi hfopen i
(zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i y -
zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i x)
hyStage with
⟨r, hr, hrproj⟩
have hq : q (y - r) = q y := by
apply (Submodule.Quotient.eq Sclosed).2
change (y - r) - y ∈ Sclosed
have hdiff : (y - r) - y = -r := by
abel
rw [hdiff]
exact Sclosed.neg_mem
(zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard_le_closed
C hC hForm psi hpsi hfopen hr)
have hproj_eq :
zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i (y - r) =
zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i x := by
rw [zcCompletedGroupAlgebraStandardAugmentationIdealProjection_sub, hrproj]
abel
have hyW : Ssys.projection i ((y - r : zcCompletedGroupAlgebraStandardAugmentationIdeal C G) :
ZCCompletedGroupAlgebra C G) ∈ W := by
have hval := congrArg Subtype.val hproj_eq
change
zcCompletedGroupAlgebraProjection C G i
((y - r : zcCompletedGroupAlgebraStandardAugmentationIdeal C G) :
ZCCompletedGroupAlgebra C G) =
zcCompletedGroupAlgebraProjection C G i
(x : ZCCompletedGroupAlgebra C G) at hval
have hxW' :
zcCompletedGroupAlgebraProjection C G i
(x : ZCCompletedGroupAlgebra C G) ∈ W := by
simpa [Ssys, zcCompletedGroupAlgebraSystem] using hxW
change
zcCompletedGroupAlgebraProjection C G i
(((y - r : zcCompletedGroupAlgebraStandardAugmentationIdeal C G) :
ZCCompletedGroupAlgebra C G)) ∈ W
rw [hval]
exact hxW'
have hyV :
(((y - r : zcCompletedGroupAlgebraStandardAugmentationIdeal C G) :
ZCCompletedGroupAlgebra C G)) ∈ V :=
hWV hyW
have hyU : q (y - r) ∈ U := by
have hyPre :
(y - r : zcCompletedGroupAlgebraStandardAugmentationIdeal C G) ∈
(Subtype.val : zcCompletedGroupAlgebraStandardAugmentationIdeal C G →
ZCCompletedGroupAlgebra C G) ⁻¹' V := hyV
rwa [hVeq] at hyPre
rwa [hq] at hyU
· letI : TopologicalSpace Q := Tind
have hprod : Continuous stageProduct :=
continuous_induced_dom
have hcoord :
Continuous (fun z : Q =>
kernelAugmentationIdealClosedQuotientStageProjection
C hC hForm psi hpsi hfopen i z) := by
simpa [stageProduct,
kernelAugmentationIdealClosedQuotientStageProjectionProduct] using
(continuous_apply i).comp hprod
have hsingle_open :
IsOpen
({kernelAugmentationIdealClosedQuotientStageProjection
C hC hForm psi hpsi hfopen i (q x)} :
Set (KernelAugmentationIdealClosedStageQuotient C hC hForm psi hpsi hfopen i)) := by
let T :=
zcCompletedGroupAlgebraOpenImageKernelAugmentationIdealMulStageStandard
C hC hForm psi hpsi hfopen i
haveI : DiscreteTopology (zcCompletedGroupAlgebraStageAugmentationIdeal C G i) := by
infer_instance
change
@IsOpen
(KernelAugmentationIdealClosedStageQuotient C hC hForm psi hpsi hfopen i)
(TopologicalSpace.coinduced T.mkQ inferInstance)
({kernelAugmentationIdealClosedQuotientStageProjection
C hC hForm psi hpsi hfopen i (q x)} :
Set (KernelAugmentationIdealClosedStageQuotient C hC hForm psi hpsi hfopen i))
rw [isOpen_coinduced]
exact isOpen_discrete _
exact
hsingle_open.preimage hcoord
· exact rfl
· intro hU
rcases isOpen_induced_iff.mp hU with ⟨V, hVopen, hVU⟩
rw [← hVU]
exact hVopen.preimage
(continuous_kernelAugmentationIdealClosedQuotientStageProjectionProduct
C hC hForm psi hpsi hfopen)Proof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□def zcCompletedGroupAlgebraOpenImageStageRingHom
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(i : ZCCompletedGroupAlgebraIndex C G) :
ZCCompletedGroupAlgebraStage C G i →+*
ZCCompletedGroupAlgebraStage C H
(zcCompletedGroupAlgebraOpenImageTargetIndex C hForm psi hpsi hfopen i) :=
MonoidAlgebra.mapDomainRingHom (ModNCompletedCoeff i.1.modulus)
(zcCompletedGroupAlgebraOpenImageQuotientMap C hC hForm psi hpsi hfopen i)
@[simp]The finite-stage source-to-open-image group-algebra map used to descend the source-stage action on the closed finite augmentation quotient to the matching target stage.
theorem zcCompletedGroupAlgebraOpenImageStageRingHom_of
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(i : ZCCompletedGroupAlgebraIndex C G)
(q : CompletedGroupAlgebraQuotientInClass G C i.2) :
zcCompletedGroupAlgebraOpenImageStageRingHom C hC hForm psi hpsi hfopen i
(MonoidAlgebra.of (ModNCompletedCoeff i.1.modulus)
(CompletedGroupAlgebraQuotientInClass G C i.2) q) =
MonoidAlgebra.of (ModNCompletedCoeff i.1.modulus)
(CompletedGroupAlgebraQuotientInClass H C
(zcCompletedGroupAlgebraOpenImageTargetIndex C hForm psi hpsi hfopen i).2)
(zcCompletedGroupAlgebraOpenImageQuotientMap
C hC hForm psi hpsi hfopen i q)The open-image stage ring homomorphism sends generators according to the finite-stage Fox coordinate formula.
Show proof
by
simp only [zcCompletedGroupAlgebraOpenImageStageRingHom, MonoidAlgebra.of_apply]
exact MonoidAlgebra.mapDomain_singleProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem zcCompletedGroupAlgebraOpenImageStageRingHom_surjective
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(i : ZCCompletedGroupAlgebraIndex C G) :
Function.Surjective
(zcCompletedGroupAlgebraOpenImageStageRingHom
C hC hForm psi hpsi hfopen i)The finite-stage source-to-open-image group-algebra map is surjective.
Show proof
by
simpa [zcCompletedGroupAlgebraOpenImageStageRingHom,
MonoidAlgebra.mapDomainRingHom_apply] using
(Finsupp.mapDomain_surjective (M := ModNCompletedCoeff i.1.modulus)
(zcCompletedGroupAlgebraOpenImageQuotientMap_surjective
C hC hForm psi hpsi hfopen i))Proof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem zcCompletedGroupAlgebraOpenImageStageRingHom_ker_smul_mem
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(i : ZCCompletedGroupAlgebraIndex C G)
(k : ZCCompletedGroupAlgebraStage C G i)
(hk : k ∈ RingHom.ker
(zcCompletedGroupAlgebraOpenImageStageRingHom
C hC hForm psi hpsi hfopen i))
(s : zcCompletedGroupAlgebraStageAugmentationIdeal C G i) :
k • s ∈
zcCompletedGroupAlgebraOpenImageKernelAugmentationIdealMulStageStandard
C hC hForm psi hpsi hfopen iSource-stage elements in the kernel of the open-image stage map multiply the finite augmentation stage into the finite kernel-product denominator.
Show proof
by
let f := zcCompletedGroupAlgebraOpenImageQuotientMap
C hC hForm psi hpsi hfopen i
let T :=
zcCompletedGroupAlgebraOpenImageKernelAugmentationIdealMulStageStandard
C hC hForm psi hpsi hfopen i
have hkIdeal :
k ∈ groupAlgebraMapDomainKernelAugmentationIdeal
(R := ModNCompletedCoeff i.1.modulus) f := by
have hk' :
k ∈ RingHom.ker
(MonoidAlgebra.mapDomainRingHom
(ModNCompletedCoeff i.1.modulus) f) := by
simpa [zcCompletedGroupAlgebraOpenImageStageRingHom, f] using hk
rwa [groupAlgebraMapDomainRingHom_ker_eq_kernelAugmentationIdeal_of_surjective
(R := ModNCompletedCoeff i.1.modulus) f
(zcCompletedGroupAlgebraOpenImageQuotientMap_surjective
C hC hForm psi hpsi hfopen i)] at hk'
change k • s ∈ T
rw [groupAlgebraMapDomainKernelAugmentationIdeal] at hkIdeal
refine Submodule.span_induction
(p := fun k _ => k • s ∈ T) ?hgen ?hzero ?hadd ?hsmul hkIdeal
· rintro _ ⟨q, rfl⟩
exact
zcCompletedGAOpenImageKernelAugmentationIdealMulStageStandard_generator_mem
C hC hForm psi hpsi hfopen i q s
· simp only [zero_smul, zero_mem]
· intro a b _ _ ha hb
simpa [add_smul] using T.add_mem ha hb
· intro a b _ hb
simpa [mul_smul] using T.smul_mem a hbProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem kernelAugmentationIdealClosedStageQuotient_openImageStageRingHom_ker_smul_eq_zero
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(i : ZCCompletedGroupAlgebraIndex C G)
(k : ZCCompletedGroupAlgebraStage C G i)
(hk : k ∈ RingHom.ker
(zcCompletedGroupAlgebraOpenImageStageRingHom
C hC hForm psi hpsi hfopen i))
(x : KernelAugmentationIdealClosedStageQuotient C hC hForm psi hpsi hfopen i) :
k • x = 0Source-stage kernels act trivially on the closed finite augmentation quotient.
Show proof
by
let T :=
zcCompletedGroupAlgebraOpenImageKernelAugmentationIdealMulStageStandard
C hC hForm psi hpsi hfopen i
refine Submodule.Quotient.induction_on (p := T) x ?_
intro s
apply (Submodule.Quotient.mk_eq_zero (p := T)).2
exact
zcCompletedGroupAlgebraOpenImageStageRingHom_ker_smul_mem
C hC hForm psi hpsi hfopen i k hk sProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□def kernelAugmentationIdealClosedStageQuotientTargetStageModule
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(i : ZCCompletedGroupAlgebraIndex C G) :
Module
(ZCCompletedGroupAlgebraStage C H
(zcCompletedGroupAlgebraOpenImageTargetIndex C hForm psi hpsi hfopen i))
(KernelAugmentationIdealClosedStageQuotient C hC hForm psi hpsi hfopen i) := by
let φ := zcCompletedGroupAlgebraOpenImageStageRingHom
C hC hForm psi hpsi hfopen i
let hφ := zcCompletedGroupAlgebraOpenImageStageRingHom_surjective
C hC hForm psi hpsi hfopen i
letI : SMul
(ZCCompletedGroupAlgebraStage C H
(zcCompletedGroupAlgebraOpenImageTargetIndex C hForm psi hpsi hfopen i))
(KernelAugmentationIdealClosedStageQuotient C hC hForm psi hpsi hfopen i) :=
⟨fun a x => Function.surjInv hφ a • x⟩
refine hφ.moduleLeft φ ?_
intro a x
change Function.surjInv hφ (φ a) • x = a • x
have hdiff : Function.surjInv hφ (φ a) - a ∈ RingHom.ker φ := by
rw [RingHom.mem_ker, map_sub, Function.surjInv_eq hφ, sub_self]
have hzero :=
kernelAugmentationIdealClosedStageQuotient_openImageStageRingHom_ker_smul_eq_zero
C hC hForm psi hpsi hfopen i
(Function.surjInv hφ (φ a) - a) hdiff x
rw [sub_smul] at hzero
exact sub_eq_zero.mp hzeroThe finite closed augmentation quotient as a module over the matching open-image target group-algebra stage.
theorem kernelAugmentationIdealClosedStageQuotientTargetStageModule_map_smul
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(i : ZCCompletedGroupAlgebraIndex C G)
(a : ZCCompletedGroupAlgebraStage C G i)
(x : KernelAugmentationIdealClosedStageQuotient C hC hForm psi hpsi hfopen i) :
letI : Module
(ZCCompletedGroupAlgebraStage C H
(zcCompletedGroupAlgebraOpenImageTargetIndex C hForm psi hpsi hfopen i))
(KernelAugmentationIdealClosedStageQuotient C hC hForm psi hpsi hfopen i)The stage quotient target module map is compatible with scalar multiplication.
Show proof
kernelAugmentationIdealClosedStageQuotientTargetStageModule
C hC hForm psi hpsi hfopen i
zcCompletedGroupAlgebraOpenImageStageRingHom C hC hForm psi hpsi hfopen i a • x =
a • x := by
let φ := zcCompletedGroupAlgebraOpenImageStageRingHom
C hC hForm psi hpsi hfopen i
let hφ := zcCompletedGroupAlgebraOpenImageStageRingHom_surjective
C hC hForm psi hpsi hfopen i
letI : Module
(ZCCompletedGroupAlgebraStage C H
(zcCompletedGroupAlgebraOpenImageTargetIndex C hForm psi hpsi hfopen i))
(KernelAugmentationIdealClosedStageQuotient C hC hForm psi hpsi hfopen i) :=
kernelAugmentationIdealClosedStageQuotientTargetStageModule
C hC hForm psi hpsi hfopen i
change Function.surjInv hφ (φ a) • x = a • x
have hdiff : Function.surjInv hφ (φ a) - a ∈ RingHom.ker φ := by
rw [RingHom.mem_ker, map_sub, Function.surjInv_eq hφ, sub_self]
have hzero :=
kernelAugmentationIdealClosedStageQuotient_openImageStageRingHom_ker_smul_eq_zero
C hC hForm psi hpsi hfopen i
(Function.surjInv hφ (φ a) - a) hdiff x
rw [sub_smul] at hzero
exact sub_eq_zero.mp hzeroProof. Unfold the relevant Fox differential, quotient, or finite-stage map. The asserted formula is checked on group-ring generators or quotient classes; linearity, the crossed-derivation rule, and the defining augmentation/kernel relations extend the calculation to the whole module.
□def zcCompletedDifferentialModuleOpenImageIndex
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(_hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(i : ZCCompletedGroupAlgebraIndex C G) :
ZCCompletedDifferentialModuleIndex C psi.toMonoidHom where
source := OrderDual.ofDual i.2
target := zcCompletedGroupAlgebraOpenImageTargetIndex C hForm psi hpsi hfopen i
compatible := by
intro g hg
change psi g ∈
((((OrderDual.ofDual
(zcCompletedGroupAlgebraOpenImageTargetIndex C hForm psi hpsi hfopen i).2).1 :
OpenNormalSubgroup H) : Subgroup H))
exact ⟨g, hg, rfl⟩The finite source quotient paired with the open-image target stage for a source group-algebra coordinate.
def kernelAugmentationIdealClosedStageQuotientBoundary
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(i : ZCCompletedGroupAlgebraIndex C G)
(q : CompletedGroupAlgebraQuotientInClass G C i.2) :
KernelAugmentationIdealClosedStageQuotient C hC hForm psi hpsi hfopen i :=
Submodule.Quotient.mk
(p := zcCompletedGroupAlgebraOpenImageKernelAugmentationIdealMulStageStandard
C hC hForm psi hpsi hfopen i)
(zcCompletedGroupAlgebraStageAugmentationGeneratorSubtype C G i q)
@[simp 900]theorem zcCompletedDifferentialModuleOpenImageIndex_stageScalar
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(i : ZCCompletedGroupAlgebraIndex C G)
(q : zcCompletedDifferentialModuleStageSource C psi.toMonoidHom
(zcCompletedDifferentialModuleOpenImageIndex C hC hForm psi hpsi hfopen i)) :
zcCompletedDifferentialModuleStageScalar C psi.toMonoidHom
(zcCompletedDifferentialModuleOpenImageIndex C hC hForm psi hpsi hfopen i) q =
zcCompletedGroupAlgebraOpenImageStageRingHom C hC hForm psi hpsi hfopen i
(MonoidAlgebra.of (ModNCompletedCoeff i.1.modulus)
(CompletedGroupAlgebraQuotientInClass G C i.2) q)The open-image stage condition for the \(\mathbb{Z}_C\)-completed differential module is equivalent to the finite-stage coordinate condition.
Show proof
by
refine QuotientGroup.induction_on q ?_
intro g
rw [zcCompletedGroupAlgebraOpenImageStageRingHom_of]
have hq :
zcCompletedGroupAlgebraOpenImageQuotientMap C hC hForm psi hpsi hfopen i
(QuotientGroup.mk'
((((OrderDual.ofDual i.2).1 : OpenNormalSubgroup G) : Subgroup G)) g) =
QuotientGroup.mk'
((((OrderDual.ofDual
(zcCompletedGroupAlgebraOpenImageTargetIndex C hForm psi hpsi hfopen i).2).1 :
OpenNormalSubgroup H) : Subgroup H)) (psi g) :=
zcCompletedGroupAlgebraOpenImageQuotientMap_mk
C hC hForm psi hpsi hfopen i g
simpa [zcCompletedDifferentialModuleStageScalar,
zcCompletedDifferentialModuleStagePsi,
zcCompletedDifferentialModuleOpenImageIndex] using
congrArg
(MonoidAlgebra.of (ModNCompletedCoeff i.1.modulus)
(CompletedGroupAlgebraQuotientInClass H C
(zcCompletedGroupAlgebraOpenImageTargetIndex C hForm psi hpsi hfopen i).2))
hq.symmProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem kernelAugmentationIdealClosedStageQuotientBoundary_isCrossedDifferential
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(i : ZCCompletedGroupAlgebraIndex C G) :
let jThe finite source-boundary coordinate is a crossed differential over the matching open-image target stage.
Show proof
zcCompletedDifferentialModuleOpenImageIndex C hC hForm psi hpsi hfopen i
letI : Module (zcCompletedDifferentialModuleStageRing C psi.toMonoidHom j)
(KernelAugmentationIdealClosedStageQuotient C hC hForm psi hpsi hfopen i) :=
kernelAugmentationIdealClosedStageQuotientTargetStageModule
C hC hForm psi hpsi hfopen i
IsCrossedDifferential
(zcCompletedDifferentialModuleStageScalar C psi.toMonoidHom j)
(fun q : zcCompletedDifferentialModuleStageSource C psi.toMonoidHom j =>
kernelAugmentationIdealClosedStageQuotientBoundary
C hC hForm psi hpsi hfopen i q) := by
let j := zcCompletedDifferentialModuleOpenImageIndex C hC hForm psi hpsi hfopen i
let T :=
zcCompletedGroupAlgebraOpenImageKernelAugmentationIdealMulStageStandard
C hC hForm psi hpsi hfopen i
letI : Module (zcCompletedDifferentialModuleStageRing C psi.toMonoidHom j)
(KernelAugmentationIdealClosedStageQuotient C hC hForm psi hpsi hfopen i) :=
kernelAugmentationIdealClosedStageQuotientTargetStageModule
C hC hForm psi hpsi hfopen i
change
IsCrossedDifferential
(zcCompletedDifferentialModuleStageScalar C psi.toMonoidHom j)
(fun q : zcCompletedDifferentialModuleStageSource C psi.toMonoidHom j =>
kernelAugmentationIdealClosedStageQuotientBoundary
C hC hForm psi hpsi hfopen i q)
intro q r
rw [zcCompletedDifferentialModuleOpenImageIndex_stageScalar]
rw [kernelAugmentationIdealClosedStageQuotientTargetStageModule_map_smul
(C := C) (hC := hC) (hForm := hForm) (psi := psi)
(hpsi := hpsi) (hfopen := hfopen) (i := i)
(a := MonoidAlgebra.of (ModNCompletedCoeff i.1.modulus)
(CompletedGroupAlgebraQuotientInClass G C i.2) q)
(x := kernelAugmentationIdealClosedStageQuotientBoundary
C hC hForm psi hpsi hfopen i r)]
change
Submodule.Quotient.mk (p := T)
(zcCompletedGroupAlgebraStageAugmentationGeneratorSubtype C G i (q * r)) =
Submodule.Quotient.mk (p := T)
(zcCompletedGroupAlgebraStageAugmentationGeneratorSubtype C G i q) +
MonoidAlgebra.of (ModNCompletedCoeff i.1.modulus)
(CompletedGroupAlgebraQuotientInClass G C i.2) q •
Submodule.Quotient.mk (p := T)
(zcCompletedGroupAlgebraStageAugmentationGeneratorSubtype C G i r)
rw [← Submodule.Quotient.mk_smul, ← Submodule.Quotient.mk_add]
apply congrArg (fun s : zcCompletedGroupAlgebraStageAugmentationIdeal C G i =>
Submodule.Quotient.mk (p := T) s)
apply Subtype.ext
change
MonoidAlgebra.of (ModNCompletedCoeff i.1.modulus)
(CompletedGroupAlgebraQuotientInClass G C i.2) (q * r) - 1 =
(MonoidAlgebra.of (ModNCompletedCoeff i.1.modulus)
(CompletedGroupAlgebraQuotientInClass G C i.2) q - 1) +
MonoidAlgebra.of (ModNCompletedCoeff i.1.modulus)
(CompletedGroupAlgebraQuotientInClass G C i.2) q *
(MonoidAlgebra.of (ModNCompletedCoeff i.1.modulus)
(CompletedGroupAlgebraQuotientInClass G C i.2) r - 1)
rw [map_mul, mul_sub, mul_one]
abelProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□def kernelAugmentationIdealClosedStageQuotientBoundaryLift
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(i : ZCCompletedGroupAlgebraIndex C G) :
let j := zcCompletedDifferentialModuleOpenImageIndex C hC hForm psi hpsi hfopen i
letI : Module (zcCompletedDifferentialModuleStageRing C psi.toMonoidHom j)
(KernelAugmentationIdealClosedStageQuotient C hC hForm psi hpsi hfopen i) :=
kernelAugmentationIdealClosedStageQuotientTargetStageModule
C hC hForm psi hpsi hfopen i
CrossedDifferentialPreModule
(zcCompletedDifferentialModuleStageRing C psi.toMonoidHom j)
(zcCompletedDifferentialModuleStageSource C psi.toMonoidHom j) →ₗ[
zcCompletedDifferentialModuleStageRing C psi.toMonoidHom j]
KernelAugmentationIdealClosedStageQuotient C hC hForm psi hpsi hfopen i := by
let j := zcCompletedDifferentialModuleOpenImageIndex C hC hForm psi hpsi hfopen i
letI : Module (zcCompletedDifferentialModuleStageRing C psi.toMonoidHom j)
(KernelAugmentationIdealClosedStageQuotient C hC hForm psi hpsi hfopen i) :=
kernelAugmentationIdealClosedStageQuotientTargetStageModule
C hC hForm psi hpsi hfopen i
exact
crossedDifferentialModuleLiftLinear
(R := zcCompletedDifferentialModuleStageRing C psi.toMonoidHom j)
(kernelAugmentationIdealClosedStageQuotientBoundary
C hC hForm psi hpsi hfopen i)
@[simp 900]theorem kernelAugmentationIdealClosedStageQuotientBoundaryLift_single
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(i : ZCCompletedGroupAlgebraIndex C G)
(q : zcCompletedDifferentialModuleStageSource C psi.toMonoidHom
(zcCompletedDifferentialModuleOpenImageIndex C hC hForm psi hpsi hfopen i))
(a : zcCompletedDifferentialModuleStageRing C psi.toMonoidHom
(zcCompletedDifferentialModuleOpenImageIndex C hC hForm psi hpsi hfopen i)) :
let jThe boundary lift lands in the finite-stage closed augmentation quotient by the Fox-differential kernel condition.
Show proof
zcCompletedDifferentialModuleOpenImageIndex C hC hForm psi hpsi hfopen i
letI : Module (zcCompletedDifferentialModuleStageRing C psi.toMonoidHom j)
(KernelAugmentationIdealClosedStageQuotient C hC hForm psi hpsi hfopen i) :=
kernelAugmentationIdealClosedStageQuotientTargetStageModule
C hC hForm psi hpsi hfopen i
kernelAugmentationIdealClosedStageQuotientBoundaryLift
C hC hForm psi hpsi hfopen i (Finsupp.single q a) =
a • kernelAugmentationIdealClosedStageQuotientBoundary
C hC hForm psi hpsi hfopen i q := by
let j := zcCompletedDifferentialModuleOpenImageIndex C hC hForm psi hpsi hfopen i
letI : Module (zcCompletedDifferentialModuleStageRing C psi.toMonoidHom j)
(KernelAugmentationIdealClosedStageQuotient C hC hForm psi hpsi hfopen i) :=
kernelAugmentationIdealClosedStageQuotientTargetStageModule
C hC hForm psi hpsi hfopen i
simp only [ContinuousMonoidHom.coe_toMonoidHom, Lean.Elab.WF.paramLet,
kernelAugmentationIdealClosedStageQuotientBoundaryLift, crossedDifferentialModuleLiftLinear_single]Proof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□def zcCompletedGroupAlgebraKernelAugmentationQuotientToClosedQuotient
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi) :
KernelAugmentationIdealQuotient C psi →ₗ[ZCCompletedGroupAlgebra C G]
KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen :=
(zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi).mapQ
(zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen)
LinearMap.id
(by
intro x hx
simpa using
zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard_le_closed
C hC hForm psi hpsi hfopen hx)
@[simp]The canonical quotient map from the algebraic kernel-product quotient to its closed finite-stage quotient.
theorem zcCompletedGroupAlgebraKernelAugmentationQuotientToClosedQuotient_mk
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G) :
zcCompletedGroupAlgebraKernelAugmentationQuotientToClosedQuotient
C hC hForm psi hpsi hfopen (Submodule.Quotient.mk x) =
Submodule.Quotient.mk xThe map from the algebraic kernel-product quotient to the closed quotient sends a representative to its closed quotient class.
Show proof
by
rw [zcCompletedGroupAlgebraKernelAugmentationQuotientToClosedQuotient,
Submodule.mapQ_apply]
rflProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem zcCompletedGAKerAugQuotToClosedQuotient_inj_iff_eq_closed
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi) :
Function.Injective
(zcCompletedGroupAlgebraKernelAugmentationQuotientToClosedQuotient
C hC hForm psi hpsi hfopen) ↔
(zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi :
Set (zcCompletedGroupAlgebraStandardAugmentationIdeal C G)) =
(zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen :
Set (zcCompletedGroupAlgebraStandardAugmentationIdeal C G))Show proof
by
let S := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi
let T :=
zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen
let Q :=
zcCompletedGroupAlgebraKernelAugmentationQuotientToClosedQuotient
C hC hForm psi hpsi hfopen
constructor
· intro hQ
apply Set.Subset.antisymm
· intro x hx
exact zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard_le_closed
C hC hForm psi hpsi hfopen hx
· intro x hx
have hmap :
Q (Submodule.Quotient.mk (p := S) x) = 0 := by
rw [zcCompletedGroupAlgebraKernelAugmentationQuotientToClosedQuotient_mk
(C := C) (hC := hC) (hForm := hForm) (psi := psi)
(hpsi := hpsi) (hfopen := hfopen) x]
exact (Submodule.Quotient.mk_eq_zero (p := T) (x := x)).2 hx
have hzero :
(Submodule.Quotient.mk (p := S) x :
KernelAugmentationIdealQuotient C psi) = 0 := by
apply hQ
rw [hmap, map_zero]
exact (Submodule.Quotient.mk_eq_zero (p := S) (x := x)).1 hzero
· intro hEq a b hxy
revert b
refine Submodule.Quotient.induction_on
(p := S) a ?_
intro x y hxy
revert hxy
refine Submodule.Quotient.induction_on
(p := S) y ?_
intro y hxy
apply (Submodule.Quotient.eq S).2
have hmemT : x - y ∈ T := by
apply (Submodule.Quotient.eq T).1
simpa [Q, zcCompletedGroupAlgebraKernelAugmentationQuotientToClosedQuotient_mk
(C := C) (hC := hC) (hForm := hForm) (psi := psi)
(hpsi := hpsi) (hfopen := hfopen)] using hxy
have hEqST : (S : Set (zcCompletedGroupAlgebraStandardAugmentationIdeal C G)) = T := by
simpa [S, T] using hEq
have hmemTset :
(x - y) ∈
(T : Set (zcCompletedGroupAlgebraStandardAugmentationIdeal C G)) := hmemT
have hmemSset :
(x - y) ∈
(S : Set (zcCompletedGroupAlgebraStandardAugmentationIdeal C G)) := by
rw [hEqST]
exact hmemTset
exact hmemSsetProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem isClosed_zcCompletedGAKernelAugmentationIdealMulStandard_iff_toClosedQuotient_inj
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi) :
IsClosed
((zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi :
Set (zcCompletedGroupAlgebraStandardAugmentationIdeal C G))) ↔
Function.Injective
(zcCompletedGroupAlgebraKernelAugmentationQuotientToClosedQuotient
C hC hForm psi hpsi hfopen)Closedness of \(I(\ker \psi)I(G)\) is equivalently the injectivity of the canonical map from the algebraic source augmentation quotient to the closed finite-stage quotient.
Show proof
by
rw [isClosed_zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard_iff_eq_closed
C hC hForm psi hpsi hfopen]
exact
(zcCompletedGAKerAugQuotToClosedQuotient_inj_iff_eq_closed
C hC hForm psi hpsi hfopen).symmProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□def zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationQuotient
(psi : ContinuousMonoidHom G H) (g : G) :
KernelAugmentationIdealQuotient C psi :=
Submodule.Quotient.mk
⟨zcCompletedGroupAlgebraBoundary C (MonoidHom.id G) g,
zcCompletedGroupAlgebraBoundary_mem_standardAugmentationIdeal
C G (MonoidHom.id G) g⟩The source Fox boundary, valued in the source augmentation quotient.
theorem zcCompletedGASourceBoundaryToKerAugQuot_isCrossedDiff
(psi : ContinuousMonoidHom G H) :
IsCrossedDifferential
(zcCompletedGroupAlgebraScalar C (MonoidHom.id G))
(zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationQuotient C psi)The source-boundary map to the source augmentation quotient is a crossed differential for the source completed group algebra.
Show proof
by
intro g h
have hboundary :=
zcCompletedGroupAlgebraBoundaryToStandardAugmentationIdeal_isCrossedDifferential
C G (MonoidHom.id G) g h
let p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi
change
Submodule.Quotient.mk (p := p)
(zcCompletedGroupAlgebraBoundaryToStandardAugmentationIdeal
C G (MonoidHom.id G) (g * h)) =
Submodule.Quotient.mk (p := p)
(zcCompletedGroupAlgebraBoundaryToStandardAugmentationIdeal
C G (MonoidHom.id G) g) +
zcCompletedGroupAlgebraScalar C (MonoidHom.id G) g •
Submodule.Quotient.mk (p := p)
(zcCompletedGroupAlgebraBoundaryToStandardAugmentationIdeal
C G (MonoidHom.id G) h)
rw [hboundary]
rw [Submodule.Quotient.mk_add, Submodule.Quotient.mk_smul]Proof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□def zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi) (g : G) :
KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen :=
Submodule.Quotient.mk
⟨zcCompletedGroupAlgebraBoundary C (MonoidHom.id G) g,
zcCompletedGroupAlgebraBoundary_mem_standardAugmentationIdeal
C G (MonoidHom.id G) g⟩
@[simp]The source Fox boundary, valued in the closed source augmentation quotient.
theorem zcCompletedGroupAlgebraKernelAugmentationQuotientToClosedQuotient_sourceBoundary
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi) (g : G) :
zcCompletedGroupAlgebraKernelAugmentationQuotientToClosedQuotient
C hC hForm psi hpsi hfopen
(zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationQuotient C psi g) =
zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
C hC hForm psi hpsi hfopen gThe map from the algebraic source augmentation quotient to the closed quotient sends source boundary classes to the corresponding closed source boundary classes.
Show proof
by
rw [zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationQuotient,
zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient,
zcCompletedGroupAlgebraKernelAugmentationQuotientToClosedQuotient_mk]Proof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem zcCompletedGASourceBoundaryToKerAugClosedQuot_isCrossedDiff
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi) :
IsCrossedDifferential
(zcCompletedGroupAlgebraScalar C (MonoidHom.id G))
(zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
C hC hForm psi hpsi hfopen)The source-boundary map to the closed source augmentation quotient is a crossed differential for the source completed group algebra.
Show proof
by
intro g h
have hboundary :=
zcCompletedGroupAlgebraBoundaryToStandardAugmentationIdeal_isCrossedDifferential
C G (MonoidHom.id G) g h
let p :=
zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen
change
Submodule.Quotient.mk (p := p)
(zcCompletedGroupAlgebraBoundaryToStandardAugmentationIdeal
C G (MonoidHom.id G) (g * h)) =
Submodule.Quotient.mk (p := p)
(zcCompletedGroupAlgebraBoundaryToStandardAugmentationIdeal
C G (MonoidHom.id G) g) +
zcCompletedGroupAlgebraScalar C (MonoidHom.id G) g •
Submodule.Quotient.mk (p := p)
(zcCompletedGroupAlgebraBoundaryToStandardAugmentationIdeal
C G (MonoidHom.id G) h)
rw [hboundary]
rw [Submodule.Quotient.mk_add, Submodule.Quotient.mk_smul]Proof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem continuous_zcCompletedGASourceBoundaryToKerAugClosedQuot
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi) :
Continuous
(zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
C hC hForm psi hpsi hfopen)The source Fox boundary into the closed source augmentation quotient is continuous.
Show proof
by
have hstd :
Continuous
(zcCompletedGroupAlgebraBoundaryToStandardAugmentationIdeal
C G (MonoidHom.id G)) := by
have hval :
Continuous (fun g : G =>
(zcCompletedGroupAlgebraBoundaryToStandardAugmentationIdeal
C G (MonoidHom.id G) g : ZCCompletedGroupAlgebra C G)) :=
continuous_zcCompletedGroupAlgebraBoundary
(C := C) (G := G) (MonoidHom.id G) continuous_id
exact Continuous.subtype_mk hval
(fun g =>
(zcCompletedGroupAlgebraBoundaryToStandardAugmentationIdeal
C G (MonoidHom.id G) g).2)
have hq :
Continuous (fun x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G =>
(Submodule.Quotient.mk x :
KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)) :=
continuous_quotient_mk'
exact hq.comp hstdProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem zcCompletedGroupAlgebraKernelAugmentationIdealQuotient_mk_generator_smul
(psi : ContinuousMonoidHom G H) (n : ProfiniteKernelSubgroup psi)
(s : zcCompletedGroupAlgebraStandardAugmentationIdeal C G) :
Submodule.Quotient.mk
(p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi)
((zcGroupLike C G n.1 - 1) • s) = 0Products \((n-1)s\) vanish in the source augmentation quotient.
Show proof
by
apply (Submodule.Quotient.mk_eq_zero
(p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi)).2
exact zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard_generator_mem C psi n sProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem zcCompletedGroupAlgebraKernelAugmentationQuotient_groupLike_smul_eq_of_map_eq
(psi : ContinuousMonoidHom G H) {g₁ g₂ : G} (h : psi g₁ = psi g₂)
(x : KernelAugmentationIdealQuotient C psi) :
zcGroupLike C G g₁ • x = zcGroupLike C G g₂ • xSource group-like actions with the same image under psi agree on the source augmentation quotient. This is the algebraic descent statement needed before a completed target scalar action can be installed.
Show proof
by
refine Submodule.Quotient.induction_on
(p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi) x ?_
intro y
apply (Submodule.Quotient.eq
(zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi)).2
let n : ProfiniteKernelSubgroup psi :=
⟨g₂⁻¹ * g₁, by
change psi (g₂⁻¹ * g₁) = 1
rw [map_mul, map_inv, h]
simp only [inv_mul_cancel]⟩
have hgen :
(zcGroupLike C G n.1 - 1) • y ∈
zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi :=
zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard_generator_mem C psi n y
have hmem :
zcGroupLike C G g₂ • ((zcGroupLike C G n.1 - 1) • y) ∈
zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi :=
(zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi).smul_mem
(zcGroupLike C G g₂) hgen
convert hmem using 1
apply Subtype.ext
change zcGroupLike C G g₁ * (y : ZCCompletedGroupAlgebra C G) -
zcGroupLike C G g₂ * (y : ZCCompletedGroupAlgebra C G) =
zcGroupLike C G g₂ *
((zcGroupLike C G n.1 - 1) * (y : ZCCompletedGroupAlgebra C G))
rw [sub_mul, one_mul, mul_sub, ← mul_assoc, ← map_mul]
simp only [mul_inv_cancel_left, n]Proof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□def zcCompletedGroupAlgebraKernelAugmentationQuotientTargetGroupLikeEndOfSurjective
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi) (h : H) :
AddMonoid.End (KernelAugmentationIdealQuotient C psi) where
toFun x := zcGroupLike C G (Function.surjInv hpsi h) • x
map_zero' := smul_zero _
map_add' := by
intro x y
rw [smul_add]The additive endomorphism of the source augmentation quotient induced by any chosen lift of a target element.
theorem zcCompletedGAKerAugQuotTargetGroupLikeEndOfSurjective_apply
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi) (h : H)
(x : KernelAugmentationIdealQuotient C psi) :
zcCompletedGroupAlgebraKernelAugmentationQuotientTargetGroupLikeEndOfSurjective
C psi hpsi h x =
zcGroupLike C G (Function.surjInv hpsi h) • xThe chosen-lift target group-like endomorphism acts by scalar multiplication with the completed group-like element of the selected source lift.
Show proof
rflProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem zcCompletedGAKerAugQuotTargetGroupLikeEndOfSurjective_eq_smul_of_map_eq
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi) {h : H} {g : G}
(hg : psi g = h) (x : KernelAugmentationIdealQuotient C psi) :
zcCompletedGroupAlgebraKernelAugmentationQuotientTargetGroupLikeEndOfSurjective
C psi hpsi h x =
zcGroupLike C G g • xThe chosen-lift target action agrees with scalar multiplication by any source lift of the same target element.
Show proof
by
have hlift : psi (Function.surjInv hpsi h) = psi g := by
rw [Function.surjInv_eq hpsi h, hg]
exact zcCompletedGroupAlgebraKernelAugmentationQuotient_groupLike_smul_eq_of_map_eq
C psi hlift xProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem zcCompletedGAKerAugQuotTargetGroupLikeEndOfSurjective_one_apply
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(x : KernelAugmentationIdealQuotient C psi) :
zcCompletedGroupAlgebraKernelAugmentationQuotientTargetGroupLikeEndOfSurjective
C psi hpsi (1 : H) x = xThe chosen-lift target group-like endomorphism for the identity target element acts as the identity.
Show proof
by
have hmap : psi (Function.surjInv hpsi (1 : H)) = psi (1 : G) := by
rw [Function.surjInv_eq hpsi (1 : H), map_one]
have hsmul :=
zcCompletedGroupAlgebraKernelAugmentationQuotient_groupLike_smul_eq_of_map_eq
C psi hmap x
simpa using hsmulProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem zcCompletedGAKerAugQuotTargetGroupLikeEndOfSurjective_mul_apply
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi) (h₁ h₂ : H)
(x : KernelAugmentationIdealQuotient C psi) :
zcCompletedGroupAlgebraKernelAugmentationQuotientTargetGroupLikeEndOfSurjective
C psi hpsi (h₁ * h₂) x =
zcCompletedGroupAlgebraKernelAugmentationQuotientTargetGroupLikeEndOfSurjective
C psi hpsi h₁
(zcCompletedGroupAlgebraKernelAugmentationQuotientTargetGroupLikeEndOfSurjective
C psi hpsi h₂ x)The chosen-lift target group-like endomorphisms multiply pointwise on the quotient.
Show proof
by
let s₁ : G := Function.surjInv hpsi h₁
let s₂ : G := Function.surjInv hpsi h₂
let s₁₂ : G := Function.surjInv hpsi (h₁ * h₂)
have hs : psi s₁₂ = psi (s₁ * s₂) := by
rw [map_mul]
simp only [Function.surjInv_eq hpsi, s₁₂, s₁, s₂]
have hsmul :=
zcCompletedGroupAlgebraKernelAugmentationQuotient_groupLike_smul_eq_of_map_eq
C psi hs x
change zcGroupLike C G s₁₂ • x =
zcGroupLike C G s₁ • (zcGroupLike C G s₂ • x)
rw [← mul_smul]
rw [← (zcGroupLike C G).map_mul s₁ s₂]
exact hsmulProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□def zcCompletedGAKerAugQuotTargetGroupLikeActionOfSurjective
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi) :
H →* AddMonoid.End (KernelAugmentationIdealQuotient C psi) where
toFun h :=
zcCompletedGroupAlgebraKernelAugmentationQuotientTargetGroupLikeEndOfSurjective
C psi hpsi h
map_one' := by
refine AddMonoidHom.ext ?_
intro x
exact
zcCompletedGAKerAugQuotTargetGroupLikeEndOfSurjective_one_apply
C psi hpsi x
map_mul' h₁ h₂ := by
refine AddMonoidHom.ext ?_
intro x
change
zcCompletedGroupAlgebraKernelAugmentationQuotientTargetGroupLikeEndOfSurjective
C psi hpsi (h₁ * h₂) x =
zcCompletedGroupAlgebraKernelAugmentationQuotientTargetGroupLikeEndOfSurjective
C psi hpsi h₁
(zcCompletedGroupAlgebraKernelAugmentationQuotientTargetGroupLikeEndOfSurjective
C psi hpsi h₂ x)
exact
zcCompletedGAKerAugQuotTargetGroupLikeEndOfSurjective_mul_apply
C psi hpsi h₁ h₂ xThe descended group-like target action on the source augmentation quotient, for a surjective \(\psi\).
theorem zcCompletedGAKerAugQuotTargetGroupLikeActionOfSurjective_apply
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi) (h : H)
(x : KernelAugmentationIdealQuotient C psi) :
zcCompletedGAKerAugQuotTargetGroupLikeActionOfSurjective
C psi hpsi h x =
zcGroupLike C G (Function.surjInv hpsi h) • xThe descended target group-like action for a surjective map acts through the selected source lift.
Show proof
rflProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□def zcCompletedGroupAlgebraKernelAugmentationQuotientTargetCoeffEnd
(psi : ContinuousMonoidHom G H) (a : ZCCoeff C) :
AddMonoid.End (KernelAugmentationIdealQuotient C psi) where
toFun x := zcCompletedGroupAlgebraCoeffMap C G a • x
map_zero' := smul_zero _
map_add' := by
intro x y
rw [smul_add]Coefficients from \(\mathbb{Z}_C\) act on the source augmentation quotient through the source completed group algebra.
theorem zcCompletedGroupAlgebraKernelAugmentationQuotientTargetCoeffEnd_apply
(psi : ContinuousMonoidHom G H) (a : ZCCoeff C)
(x : KernelAugmentationIdealQuotient C psi) :
zcCompletedGroupAlgebraKernelAugmentationQuotientTargetCoeffEnd C psi a x =
zcCompletedGroupAlgebraCoeffMap C G a • xThe augmentation map is evaluated by projecting to the corresponding finite group-algebra stage and summing coefficients via the usual group-algebra augmentation.
Show proof
rflProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□def zcCompletedGroupAlgebraKernelAugmentationQuotientTargetCoeffAction
(psi : ContinuousMonoidHom G H) :
ZCCoeff C →+* AddMonoid.End (KernelAugmentationIdealQuotient C psi) where
toFun a := zcCompletedGroupAlgebraKernelAugmentationQuotientTargetCoeffEnd C psi a
map_zero' := by
refine AddMonoidHom.ext ?_
intro x
change zcCompletedGroupAlgebraCoeffMap C G (0 : ZCCoeff C) • x = 0
rw [map_zero, zero_smul]
map_one' := by
refine AddMonoidHom.ext ?_
intro x
change zcCompletedGroupAlgebraCoeffMap C G (1 : ZCCoeff C) • x = x
rw [map_one, one_smul]
map_add' a b := by
refine AddMonoidHom.ext ?_
intro x
change zcCompletedGroupAlgebraCoeffMap C G (a + b) • x =
zcCompletedGroupAlgebraCoeffMap C G a • x +
zcCompletedGroupAlgebraCoeffMap C G b • x
rw [map_add, add_smul]
map_mul' a b := by
refine AddMonoidHom.ext ?_
intro x
change zcCompletedGroupAlgebraCoeffMap C G (a * b) • x =
zcCompletedGroupAlgebraCoeffMap C G a •
(zcCompletedGroupAlgebraCoeffMap C G b • x)
rw [map_mul, mul_smul]The coefficient action of \(\mathbb{Z}_C\) on the source augmentation quotient.
theorem zcCompletedGroupAlgebraCoeffMap_mul_groupLike_eq_groupLike_mul_coeffMap
(a : ZCCoeff C) (g : G) :
zcCompletedGroupAlgebraCoeffMap C G a * zcGroupLike C G g =
zcGroupLike C G g * zcCompletedGroupAlgebraCoeffMap C G aCoefficient elements are central with respect to group-like elements in \(\mathbb{Z}_C\llbracket G\rrbracket\).
Show proof
by
apply Subtype.ext
funext i
letI : Fact (0 < i.1.modulus) := ⟨i.1.positive⟩
change
zcCompletedGroupAlgebraProjection C G i
(zcCompletedGroupAlgebraCoeffMap C G a * zcGroupLike C G g) =
zcCompletedGroupAlgebraProjection C G i
(zcGroupLike C G g * zcCompletedGroupAlgebraCoeffMap C G a)
rw [zcCompletedGroupAlgebraProjection_mul, zcCompletedGroupAlgebraProjection_coeffMap,
zcCompletedGroupAlgebraProjection_groupLike]
simp only [MonoidAlgebra.of, MonoidHom.coe_mk, OneHom.coe_mk, MonoidAlgebra.single_mul_single, one_mul,
mul_one, zcCompletedGroupAlgebraProjection_mul, zcCompletedGroupAlgebraProjection_groupLike,
zcCompletedGroupAlgebraProjection_coeffMap]Proof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□def kerAugQuotTargetGAActionOfSurj
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi) :
MonoidAlgebra (ZCCoeff C) H →+*
AddMonoid.End (KernelAugmentationIdealQuotient C psi) :=
MonoidAlgebra.liftNCRingHom
(zcCompletedGroupAlgebraKernelAugmentationQuotientTargetCoeffAction C psi)
(zcCompletedGAKerAugQuotTargetGroupLikeActionOfSurjective
C psi hpsi)
(by
intro a h
rw [Commute]
apply AddMonoidHom.ext
intro x
change
zcCompletedGroupAlgebraCoeffMap C G a •
(zcGroupLike C G (Function.surjInv hpsi h) • x) =
zcGroupLike C G (Function.surjInv hpsi h) •
(zcCompletedGroupAlgebraCoeffMap C G a • x)
rw [← mul_smul, ← mul_smul,
zcCompletedGroupAlgebraCoeffMap_mul_groupLike_eq_groupLike_mul_coeffMap])The algebraic target group algebra \(\mathbb{Z}_C[H]\) acts on the source augmentation quotient. This is the dense algebraic part of the eventual completed \(\mathbb{Z}_C\llbracket H\rrbracket\) scalar action.
theorem kerAugQuotTargetGAActionOfSurj_of
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi) (h : H) :
kerAugQuotTargetGAActionOfSurj
C psi hpsi (MonoidAlgebra.of (ZCCoeff C) H h) =
zcCompletedGAKerAugQuotTargetGroupLikeActionOfSurjective
C psi hpsi hThe induced target group-algebra action evaluates on group-like elements by lifting through the chosen surjection.
Show proof
by
apply AddMonoidHom.ext
intro x
simp only [kerAugQuotTargetGAActionOfSurj, MonoidAlgebra.of_apply, MonoidAlgebra.liftNCRingHom_single,
map_one, one_mul]Proof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□noncomputable abbrev
zcCompletedGroupAlgebraKernelAugmentationQuotientTargetGroupAlgebraModuleOfSurjective
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi) :
Module (MonoidAlgebra (ZCCoeff C) H) (KernelAugmentationIdealQuotient C psi) :=
Module.compHom (KernelAugmentationIdealQuotient C psi)
(kerAugQuotTargetGAActionOfSurj
C psi hpsi)The auxiliary Fox-coordinate identity is verified after projection to the corresponding finite quotient and coefficient stage.
theorem kerAugQuotTargetGAModuleOfSurj_smul
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(a : MonoidAlgebra (ZCCoeff C) H) (x : KernelAugmentationIdealQuotient C psi) :
letI : Module (MonoidAlgebra (ZCCoeff C) H) (KernelAugmentationIdealQuotient C psi)The target group-algebra module structure acts by the induced quotient action.
Show proof
zcCompletedGroupAlgebraKernelAugmentationQuotientTargetGroupAlgebraModuleOfSurjective
C psi hpsi
a • x =
kerAugQuotTargetGAActionOfSurj
C psi hpsi a x := by
rflProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem kerAugQuotTargetGAModuleOfSurj_of_smul
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi) (h : H)
(x : KernelAugmentationIdealQuotient C psi) :
letI : Module (MonoidAlgebra (ZCCoeff C) H) (KernelAugmentationIdealQuotient C psi)A group-like basis element acts on the target quotient module by the induced quotient action.
Show proof
zcCompletedGroupAlgebraKernelAugmentationQuotientTargetGroupAlgebraModuleOfSurjective
C psi hpsi
MonoidAlgebra.of (ZCCoeff C) H h • x =
zcGroupLike C G (Function.surjInv hpsi h) • x := by
letI : Module (MonoidAlgebra (ZCCoeff C) H) (KernelAugmentationIdealQuotient C psi) :=
zcCompletedGroupAlgebraKernelAugmentationQuotientTargetGroupAlgebraModuleOfSurjective
C psi hpsi
change
kerAugQuotTargetGAActionOfSurj
C psi hpsi (MonoidAlgebra.of (ZCCoeff C) H h) x =
zcGroupLike C G (Function.surjInv hpsi h) • x
rw [kerAugQuotTargetGAActionOfSurj_of]
rflProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem kerAugQuotTargetGAModuleOfSurj_of_smul_eq_source_groupLike_smul
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi) {h : H} {g : G}
(hg : psi g = h) (x : KernelAugmentationIdealQuotient C psi) :
letI : Module (MonoidAlgebra (ZCCoeff C) H) (KernelAugmentationIdealQuotient C psi)The algebraic target group-algebra action by \([h]\) agrees with source multiplication by any lift of \(h\).
Show proof
zcCompletedGroupAlgebraKernelAugmentationQuotientTargetGroupAlgebraModuleOfSurjective
C psi hpsi
MonoidAlgebra.of (ZCCoeff C) H h • x = zcGroupLike C G g • x := by
letI : Module (MonoidAlgebra (ZCCoeff C) H) (KernelAugmentationIdealQuotient C psi) :=
zcCompletedGroupAlgebraKernelAugmentationQuotientTargetGroupAlgebraModuleOfSurjective
C psi hpsi
rw [kerAugQuotTargetGAModuleOfSurj_of_smul]
have hlift : psi (Function.surjInv hpsi h) = psi g := by
rw [Function.surjInv_eq hpsi h, hg]
exact zcCompletedGroupAlgebraKernelAugmentationQuotient_groupLike_smul_eq_of_map_eq
C psi hlift xProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem zcCompletedGASourceBoundaryToKerAugQuot_isTargetGACrossedDiff_of_surj
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi) :
letI : Module (MonoidAlgebra (ZCCoeff C) H) (KernelAugmentationIdealQuotient C psi)The source boundary is a crossed differential for the descended algebraic target group-algebra coefficients.
Show proof
zcCompletedGroupAlgebraKernelAugmentationQuotientTargetGroupAlgebraModuleOfSurjective
C psi hpsi
IsCrossedDifferential
((MonoidAlgebra.of (ZCCoeff C) H).comp psi.toMonoidHom)
(zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationQuotient C psi) := by
letI : Module (MonoidAlgebra (ZCCoeff C) H) (KernelAugmentationIdealQuotient C psi) :=
zcCompletedGroupAlgebraKernelAugmentationQuotientTargetGroupAlgebraModuleOfSurjective
C psi hpsi
intro g h
have hsource :=
zcCompletedGASourceBoundaryToKerAugQuot_isCrossedDiff
C psi g h
rw [hsource]
congr 1
exact
(kerAugQuotTargetGAModuleOfSurj_of_smul_eq_source_groupLike_smul
C psi hpsi (h := psi g) (g := g) rfl
(zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationQuotient C psi h)).symmProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□def zcAlgebraicDifferentialModuleToKernelAugmentationQuotientOfSurjective
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi) :
letI : Module (MonoidAlgebra (ZCCoeff C) H) (KernelAugmentationIdealQuotient C psi) :=
zcCompletedGroupAlgebraKernelAugmentationQuotientTargetGroupAlgebraModuleOfSurjective
C psi hpsi
CrossedDifferentialModule ((MonoidAlgebra.of (ZCCoeff C) H).comp psi.toMonoidHom) →ₗ[
MonoidAlgebra (ZCCoeff C) H] KernelAugmentationIdealQuotient C psi := by
letI : Module (MonoidAlgebra (ZCCoeff C) H) (KernelAugmentationIdealQuotient C psi) :=
zcCompletedGroupAlgebraKernelAugmentationQuotientTargetGroupAlgebraModuleOfSurjective
C psi hpsi
exact
crossedDifferentialModuleLift
(A := KernelAugmentationIdealQuotient C psi)
((MonoidAlgebra.of (ZCCoeff C) H).comp psi.toMonoidHom)
(zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationQuotient C psi)
(zcCompletedGASourceBoundaryToKerAugQuot_isTargetGACrossedDiff_of_surj
C psi hpsi)The algebraic target-coefficient universal differential module maps to the source augmentation quotient by \(dg \mapsto\) \([g]-1\).
theorem zcAlgebraicDifferentialModuleToKernelAugmentationQuotientOfSurjective_universal
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi) (g : G) :
letI : Module (MonoidAlgebra (ZCCoeff C) H) (KernelAugmentationIdealQuotient C psi)The algebraic differential module maps universally to the kernel-augmentation quotient in the surjective case.
Show proof
zcCompletedGroupAlgebraKernelAugmentationQuotientTargetGroupAlgebraModuleOfSurjective
C psi hpsi
zcAlgebraicDifferentialModuleToKernelAugmentationQuotientOfSurjective C psi hpsi
(universalCrossedDifferential
((MonoidAlgebra.of (ZCCoeff C) H).comp psi.toMonoidHom) g) =
zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationQuotient C psi g := by
letI : Module (MonoidAlgebra (ZCCoeff C) H) (KernelAugmentationIdealQuotient C psi) :=
zcCompletedGroupAlgebraKernelAugmentationQuotientTargetGroupAlgebraModuleOfSurjective
C psi hpsi
exact
crossedDifferentialModuleLift_universal
(A := KernelAugmentationIdealQuotient C psi)
((MonoidAlgebra.of (ZCCoeff C) H).comp psi.toMonoidHom)
(zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationQuotient C psi)
(zcCompletedGASourceBoundaryToKerAugQuot_isTargetGACrossedDiff_of_surj
C psi hpsi) gProof. Unfold the relevant Fox differential, quotient, or finite-stage map. The asserted formula is checked on group-ring generators or quotient classes; linearity, the crossed-derivation rule, and the defining augmentation/kernel relations extend the calculation to the whole module.
□theorem mulStandard_mul_mem_of_mem_kernelAugIdealMul
(psi : ContinuousMonoidHom G H)
{k : ZCCompletedGroupAlgebra C G}
(hk : k ∈ zcCompletedGroupAlgebraKernelAugmentationIdealMul C psi)
(y : zcCompletedGroupAlgebraStandardAugmentationIdeal C G) :
(⟨k * (y : ZCCompletedGroupAlgebra C G),
(zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left k y.2⟩ :
zcCompletedGroupAlgebraStandardAugmentationIdeal C G) ∈
zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psiMultiplying an algebraic kernel-augmentation element by a standard augmentation element lands in the algebraic product \(I(\ker \psi)I(G)\). The remaining completed-target descent problem is exactly replacing the first hypothesis by membership in the completed map kernel.
Show proof
by
let S := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi
refine Submodule.span_induction
(p := fun k _ =>
∀ y : zcCompletedGroupAlgebraStandardAugmentationIdeal C G,
(⟨k * (y : ZCCompletedGroupAlgebra C G),
(zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left k y.2⟩ :
zcCompletedGroupAlgebraStandardAugmentationIdeal C G) ∈ S) ?_ ?_ ?_ ?_ hk y
· rintro _ ⟨n, rfl⟩ y
exact zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard_generator_mem C psi n y
· intro y
convert S.zero_mem using 1
ext
simp only [zero_mul, zcCompletedGroupAlgebraProjection_zero, Finsupp.coe_zero, Pi.zero_apply,
ZeroMemClass.coe_zero]
· intro a b _ _ ha hb y
have hsum : (⟨a * (y : ZCCompletedGroupAlgebra C G),
(zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left a y.2⟩ :
zcCompletedGroupAlgebraStandardAugmentationIdeal C G) +
(⟨b * (y : ZCCompletedGroupAlgebra C G),
(zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left b y.2⟩ :
zcCompletedGroupAlgebraStandardAugmentationIdeal C G) ∈ S :=
S.add_mem (ha y) (hb y)
convert hsum using 1
ext
simp only [add_mul, zcCompletedGroupAlgebraProjection_add, zcCompletedGroupAlgebraProjection_mul,
MonoidAlgebra.coe_add, Pi.add_apply, AddMemClass.mk_add_mk]
· intro a b _ hb y
have hsmul : a •
(⟨b * (y : ZCCompletedGroupAlgebra C G),
(zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left b y.2⟩ :
zcCompletedGroupAlgebraStandardAugmentationIdeal C G) ∈ S :=
S.smul_mem a (hb y)
convert hsmul using 1
ext
simp only [smul_eq_mul, mul_assoc, zcCompletedGroupAlgebraProjection_mul, SetLike.mk_smul_mk]Proof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem zcCompletedGAKernelAugmentationIdealMulStandard_mul_mem_closure_of_mem_ker_map
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
{k : ZCCompletedGroupAlgebra C G}
(hk : k ∈ RingHom.ker (zcCompletedGroupAlgebraMap C hC psi))
(y : zcCompletedGroupAlgebraStandardAugmentationIdeal C G) :
(⟨k * (y : ZCCompletedGroupAlgebra C G),
(zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left k y.2⟩ :
zcCompletedGroupAlgebraStandardAugmentationIdeal C G) ∈
closure
((zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi :
Set (zcCompletedGroupAlgebraStandardAugmentationIdeal C G)))Under the finite-stage open-map kernel theorem, a completed-kernel scalar times a standard augmentation element lands in the closure of the algebraic product. This is the strongest statement available without proving that the product submodule is closed.
Show proof
by
let R := ZCCompletedGroupAlgebra C G
let I := zcCompletedGroupAlgebraKernelAugmentationIdealMul C psi
let S := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi
let f : R → zcCompletedGroupAlgebraStandardAugmentationIdeal C G := fun a =>
⟨a * (y : R), (zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left a y.2⟩
have hf : Continuous f := by
have hmul : Continuous (fun a : R => a * (y : R)) :=
continuous_id.mul continuous_const
exact Continuous.subtype_mk hmul
(fun a => (zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left a y.2)
have hkClosure : k ∈ closure ((I : Set R)) := by
have hEq :=
closure_zcCompletedGAKernelAugmentationIdealMul_eq_ker_map_of_openMap_surj
C hC hForm psi hpsi hfopen
rw [hEq]
exact hk
have hmemImage : f k ∈ f '' closure ((I : Set R)) := ⟨k, hkClosure, rfl⟩
have hclosureImage : f k ∈ closure (f '' ((I : Set R))) :=
image_closure_subset_closure_image hf hmemImage
have himage_subset : f '' ((I : Set R)) ⊆ (S : Set _) := by
rintro _ ⟨a, ha, rfl⟩
exact
mulStandard_mul_mem_of_mem_kernelAugIdealMul
C psi ha y
exact closure_mono himage_subset hclosureImageProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem zcCompletedGAKernelAugmentationIdealMulStandard_mul_mem_closed_of_mem_ker_map
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
{k : ZCCompletedGroupAlgebra C G}
(hk : k ∈ RingHom.ker (zcCompletedGroupAlgebraMap C hC psi))
(y : zcCompletedGroupAlgebraStandardAugmentationIdeal C G) :
(⟨k * (y : ZCCompletedGroupAlgebra C G),
(zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left k y.2⟩ :
zcCompletedGroupAlgebraStandardAugmentationIdeal C G) ∈
zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopenIf \(k\) lies in the kernel of the completed group-algebra map, then \(k y\) lies in the finite-stage closed hull of \(I(\ker\psi)I(G)\) for every standard augmentation element \(y\).
Show proof
by
have hclosure :=
zcCompletedGAKernelAugmentationIdealMulStandard_mul_mem_closure_of_mem_ker_map
C hC hForm psi hpsi hfopen hk y
rwa [closure_zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard_eq_closed
C hC hForm psi hpsi hfopen] at hclosuretheorem zcCompletedGAKernelAugmentationIdealMulStandard_kernelMulStandard_le_closed
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi) :
∀ k : ZCCompletedGroupAlgebra C G,
k ∈ RingHom.ker (zcCompletedGroupAlgebraMap C hC psi) →
∀ y : zcCompletedGroupAlgebraStandardAugmentationIdeal C G,
(⟨k * (y : ZCCompletedGroupAlgebra C G),
(zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left k y.2⟩ :
zcCompletedGroupAlgebraStandardAugmentationIdeal C G) ∈
zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopenCompleted-kernel scalars send the standard source augmentation ideal into the finite-stage closed hull of \(I(\ker \psi)I(G)\).
Show proof
by
intro k hk y
exact
zcCompletedGAKernelAugmentationIdealMulStandard_mul_mem_closed_of_mem_ker_map
C hC hForm psi hpsi hfopen hk yProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem zcCompletedGAKernelAugmentationIdealMulStandard_kernelMulStandard_le_of_isClosed
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(hclosed :
IsClosed
((zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi :
Set (zcCompletedGroupAlgebraStandardAugmentationIdeal C G)))) :
∀ k : ZCCompletedGroupAlgebra C G,
k ∈ RingHom.ker (zcCompletedGroupAlgebraMap C hC psi) →
∀ y : zcCompletedGroupAlgebraStandardAugmentationIdeal C G,
(⟨k * (y : ZCCompletedGroupAlgebra C G),
(zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left k y.2⟩ :
zcCompletedGroupAlgebraStandardAugmentationIdeal C G) ∈
zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psiIf the algebraic product \(I(\ker \psi)I(G)\) is closed in the standard augmentation ideal, then completed-kernel scalars send standard augmentation elements into that product.
Show proof
by
intro k hk y
have hclosure :=
zcCompletedGAKernelAugmentationIdealMulStandard_mul_mem_closure_of_mem_ker_map
C hC hForm psi hpsi hfopen hk y
rwa [hclosed.closure_eq] at hclosureProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem kernelMulStandard_le_of_toClosedQuotient_inj
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(hinj :
Function.Injective
(zcCompletedGroupAlgebraKernelAugmentationQuotientToClosedQuotient
C hC hForm psi hpsi hfopen)) :
∀ k : ZCCompletedGroupAlgebra C G,
k ∈ RingHom.ker (zcCompletedGroupAlgebraMap C hC psi) →
∀ y : zcCompletedGroupAlgebraStandardAugmentationIdeal C G,
(⟨k * (y : ZCCompletedGroupAlgebra C G),
(zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left k y.2⟩ :
zcCompletedGroupAlgebraStandardAugmentationIdeal C G) ∈
zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psiIf the canonical map from the algebraic source augmentation quotient to the closed finite-stage quotient is injective, then completed-kernel scalars multiply standard augmentation elements into the algebraic product \(I(\ker \psi)I(G)\). This descent step uses finite-stage closed membership and converts it back to algebraic membership through injectivity of the quotient comparison map.
Show proof
by
intro k hk y
let S := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi
let T :=
zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen
let x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G :=
⟨k * (y : ZCCompletedGroupAlgebra C G),
(zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left k y.2⟩
have hxT : x ∈ T :=
zcCompletedGAKernelAugmentationIdealMulStandard_mul_mem_closed_of_mem_ker_map
C hC hForm psi hpsi hfopen hk y
have hxmap :
zcCompletedGroupAlgebraKernelAugmentationQuotientToClosedQuotient
C hC hForm psi hpsi hfopen
(Submodule.Quotient.mk (p := S) x) = 0 := by
rw [zcCompletedGroupAlgebraKernelAugmentationQuotientToClosedQuotient_mk
(C := C) (hC := hC) (hForm := hForm) (psi := psi)
(hpsi := hpsi) (hfopen := hfopen) x]
exact (Submodule.Quotient.mk_eq_zero (p := T) (x := x)).2 hxT
have hxzero :
(Submodule.Quotient.mk (p := S) x :
KernelAugmentationIdealQuotient C psi) = 0 := by
apply hinj
rw [hxmap, map_zero]
exact (Submodule.Quotient.mk_eq_zero (p := S) (x := x)).1 hxzeroProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem zcCompletedGAKerAugQuot_ker_map_smul_eq_zero_of_kernelMulStandard_le
(hC : ProCGroups.FiniteGroupClass.Hereditary C) (psi : ContinuousMonoidHom G H)
(hker_mul :
∀ k : ZCCompletedGroupAlgebra C G,
k ∈ RingHom.ker (zcCompletedGroupAlgebraMap C hC psi) →
∀ y : zcCompletedGroupAlgebraStandardAugmentationIdeal C G,
(⟨k * (y : ZCCompletedGroupAlgebra C G),
(zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left k y.2⟩ :
zcCompletedGroupAlgebraStandardAugmentationIdeal C G) ∈
zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi)
(k : ZCCompletedGroupAlgebra C G)
(hk : k ∈ RingHom.ker (zcCompletedGroupAlgebraMap C hC psi))
(x : KernelAugmentationIdealQuotient C psi) :
k • x = 0If every completed-kernel scalar sends the standard source augmentation ideal into \(I(\ker \psi)I(G)\), then the completed kernel acts trivially on the quotient.
Show proof
by
refine Submodule.Quotient.induction_on
(p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi) x ?_
intro y
apply (Submodule.Quotient.mk_eq_zero
(p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi)).2
change
(⟨k * (y : ZCCompletedGroupAlgebra C G),
(zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left k y.2⟩ :
zcCompletedGroupAlgebraStandardAugmentationIdeal C G) ∈
zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi
exact hker_mul k hk yProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□def zcCompletedGroupAlgebraTargetLiftOfSurjective
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi) :
ZCCompletedGroupAlgebra C H → ZCCompletedGroupAlgebra C G :=
Function.surjInv
(zcCompletedGroupAlgebraMap_surjective_of_surjective
(C := C) (hC := hC) hForm psi hpsi)
@[simp 900]Conditional descent of the source action to a completed target \(\mathbb{Z}_C\llbracket H\rrbracket\)-module. The extra hypothesis is exactly the missing kernel-product statement; it is kept explicit so that closure membership is not used as algebraic equality.
theorem zcCompletedGroupAlgebraMap_targetLiftOfSurjective
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(a : ZCCompletedGroupAlgebra C H) :
zcCompletedGroupAlgebraMap C hC psi
(zcCompletedGroupAlgebraTargetLiftOfSurjective C hC hForm psi hpsi a) = aCoefficient change is performed stagewise: each coefficient is transported by the given ring homomorphism while the finite quotient support is left unchanged.
Show proof
Function.surjInv_eq
(zcCompletedGroupAlgebraMap_surjective_of_surjective
(C := C) (hC := hC) hForm psi hpsi) aProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□def zcCompletedGAKerAugQuotTargetCompletedModuleOfSurjective_of_kernelMulStandard_le
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hker_mul :
∀ k : ZCCompletedGroupAlgebra C G,
k ∈ RingHom.ker (zcCompletedGroupAlgebraMap C hC psi) →
∀ y : zcCompletedGroupAlgebraStandardAugmentationIdeal C G,
(⟨k * (y : ZCCompletedGroupAlgebra C G),
(zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left k y.2⟩ :
zcCompletedGroupAlgebraStandardAugmentationIdeal C G) ∈
zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi) :
Module (ZCCompletedGroupAlgebra C H) (KernelAugmentationIdealQuotient C psi) := by
letI : SMul (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealQuotient C psi) :=
⟨fun a x => zcCompletedGroupAlgebraTargetLiftOfSurjective
C hC hForm psi hpsi a • x⟩
refine (zcCompletedGroupAlgebraMap_surjective_of_surjective
(C := C) (hC := hC) hForm psi hpsi).moduleLeft
(zcCompletedGroupAlgebraMap C hC psi) ?_
intro a x
change zcCompletedGroupAlgebraTargetLiftOfSurjective C hC hForm psi hpsi
(zcCompletedGroupAlgebraMap C hC psi a) • x =
a • x
have hdiff :
zcCompletedGroupAlgebraTargetLiftOfSurjective C hC hForm psi hpsi
(zcCompletedGroupAlgebraMap C hC psi a) - a ∈
RingHom.ker (zcCompletedGroupAlgebraMap C hC psi) := by
change zcCompletedGroupAlgebraMap C hC psi
(zcCompletedGroupAlgebraTargetLiftOfSurjective C hC hForm psi hpsi
(zcCompletedGroupAlgebraMap C hC psi a) - a) = 0
rw [map_sub, zcCompletedGroupAlgebraMap_targetLiftOfSurjective, sub_self]
have hzero :=
zcCompletedGAKerAugQuot_ker_map_smul_eq_zero_of_kernelMulStandard_le
C hC psi hker_mul
(zcCompletedGroupAlgebraTargetLiftOfSurjective C hC hForm psi hpsi
(zcCompletedGroupAlgebraMap C hC psi a) - a) hdiff x
rw [sub_smul] at hzero
exact sub_eq_zero.mp hzeroSurjectivity of the induced map is obtained from surjectivity on the underlying quotient or dense algebraic model, together with closedness of the image in the completed target.
def zcCompletedGAKerAugQuotTargetCompletedModuleOfSurjective_of_closed_kernelMulStandard
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(hclosed :
IsClosed
((zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi :
Set (zcCompletedGroupAlgebraStandardAugmentationIdeal C G)))) :
Module (ZCCompletedGroupAlgebra C H) (KernelAugmentationIdealQuotient C psi) :=
zcCompletedGAKerAugQuotTargetCompletedModuleOfSurjective_of_kernelMulStandard_le
C hC hForm psi hpsi
(zcCompletedGAKernelAugmentationIdealMulStandard_kernelMulStandard_le_of_isClosed
C hC hForm psi hpsi hfopen hclosed)Surjectivity of the induced map is obtained from surjectivity on the underlying quotient or dense algebraic model, together with closedness of the image in the completed target.
theorem zcCompletedGroupAlgebraKernelAugmentationClosedQuotient_ker_map_smul_eq_zero
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(k : ZCCompletedGroupAlgebra C G)
(hk : k ∈ RingHom.ker (zcCompletedGroupAlgebraMap C hC psi))
(x : KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :
k • x = 0The completed kernel acts trivially on the closed source augmentation quotient.
Show proof
by
refine Submodule.Quotient.induction_on
(p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen) x ?_
intro y
apply (Submodule.Quotient.mk_eq_zero
(p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen)).2
change
(⟨k * (y : ZCCompletedGroupAlgebra C G),
(zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left k y.2⟩ :
zcCompletedGroupAlgebraStandardAugmentationIdeal C G) ∈
zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen
exact
zcCompletedGAKernelAugmentationIdealMulStandard_kernelMulStandard_le_closed
C hC hForm psi hpsi hfopen k hk yProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□def kerAugClosedQuotTargetCompletedModuleOfSurj
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi) :
Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) := by
letI : SMul (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
⟨fun a x => zcCompletedGroupAlgebraTargetLiftOfSurjective
C hC hForm psi hpsi a • x⟩
refine (zcCompletedGroupAlgebraMap_surjective_of_surjective
(C := C) (hC := hC) hForm psi hpsi).moduleLeft
(zcCompletedGroupAlgebraMap C hC psi) ?_
intro a x
change zcCompletedGroupAlgebraTargetLiftOfSurjective C hC hForm psi hpsi
(zcCompletedGroupAlgebraMap C hC psi a) • x =
a • x
have hdiff :
zcCompletedGroupAlgebraTargetLiftOfSurjective C hC hForm psi hpsi
(zcCompletedGroupAlgebraMap C hC psi a) - a ∈
RingHom.ker (zcCompletedGroupAlgebraMap C hC psi) := by
change zcCompletedGroupAlgebraMap C hC psi
(zcCompletedGroupAlgebraTargetLiftOfSurjective C hC hForm psi hpsi
(zcCompletedGroupAlgebraMap C hC psi a) - a) = 0
rw [map_sub, zcCompletedGroupAlgebraMap_targetLiftOfSurjective, sub_self]
have hzero :=
zcCompletedGroupAlgebraKernelAugmentationClosedQuotient_ker_map_smul_eq_zero
C hC hForm psi hpsi hfopen
(zcCompletedGroupAlgebraTargetLiftOfSurjective C hC hForm psi hpsi
(zcCompletedGroupAlgebraMap C hC psi a) - a) hdiff x
rw [sub_smul] at hzero
exact sub_eq_zero.mp hzeroUnconditional descent of the source action to a completed target \(\mathbb{Z}_C\llbracket H\rrbracket\)-module on the closed source augmentation quotient.
theorem kerAugClosedQuotTargetCompletedModuleOfSurj_map_smul
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(a : ZCCompletedGroupAlgebra C G)
(x : KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)The closed quotient target module action is compatible with mapped scalars.
Show proof
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
zcCompletedGroupAlgebraMap C hC psi a • x = a • x := by
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
change zcCompletedGroupAlgebraTargetLiftOfSurjective C hC hForm psi hpsi
(zcCompletedGroupAlgebraMap C hC psi a) • x =
a • x
have hdiff :
zcCompletedGroupAlgebraTargetLiftOfSurjective C hC hForm psi hpsi
(zcCompletedGroupAlgebraMap C hC psi a) - a ∈
RingHom.ker (zcCompletedGroupAlgebraMap C hC psi) := by
change zcCompletedGroupAlgebraMap C hC psi
(zcCompletedGroupAlgebraTargetLiftOfSurjective C hC hForm psi hpsi
(zcCompletedGroupAlgebraMap C hC psi a) - a) = 0
rw [map_sub, zcCompletedGroupAlgebraMap_targetLiftOfSurjective, sub_self]
have hzero :=
zcCompletedGroupAlgebraKernelAugmentationClosedQuotient_ker_map_smul_eq_zero
C hC hForm psi hpsi hfopen
(zcCompletedGroupAlgebraTargetLiftOfSurjective C hC hForm psi hpsi
(zcCompletedGroupAlgebraMap C hC psi a) - a) hdiff x
rw [sub_smul] at hzero
exact sub_eq_zero.mp hzeroProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem continuous_zcCompletedGAKerAugClosedQuot_source_smul_const
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(x : KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :
Continuous (fun a : ZCCompletedGroupAlgebra C G => a • x)Source scalar multiplication on the closed source augmentation quotient is continuous in the source scalar, for a fixed quotient element.
Show proof
by
refine Submodule.Quotient.induction_on
(p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen) x ?_
intro y
have hpre :
Continuous (fun a : ZCCompletedGroupAlgebra C G =>
(⟨a * (y : ZCCompletedGroupAlgebra C G),
(zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left a y.2⟩ :
zcCompletedGroupAlgebraStandardAugmentationIdeal C G)) := by
have hmul : Continuous (fun a : ZCCompletedGroupAlgebra C G =>
a * (y : ZCCompletedGroupAlgebra C G)) :=
continuous_id.mul continuous_const
exact Continuous.subtype_mk hmul
(fun a => (zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left a y.2)
have hq :
Continuous (fun z : zcCompletedGroupAlgebraStandardAugmentationIdeal C G =>
(Submodule.Quotient.mk
(p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen) z :
KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)) :=
continuous_quotient_mk'
simpa using hq.comp hpreProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem continuous_kerAugClosedQuotTargetCompletedModuleOfSurj_smul_const
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(x : KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)Descended target scalar multiplication on the closed source augmentation quotient is continuous in the target scalar, for a fixed quotient element.
Show proof
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
Continuous (fun a : ZCCompletedGroupAlgebra C H => a • x) := by
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
let q := zcCompletedGroupAlgebraMap C hC psi
have hq : Topology.IsQuotientMap q :=
isQuotientMap_zcCompletedGroupAlgebraMap_of_surjective C hC hForm psi hpsi
rw [hq.continuous_iff]
change Continuous (fun a : ZCCompletedGroupAlgebra C G => q a • x)
have hsource :=
continuous_zcCompletedGAKerAugClosedQuot_source_smul_const
C hC hForm psi hpsi hfopen x
have hEq :
(fun a : ZCCompletedGroupAlgebra C G => q a • x) =
(fun a : ZCCompletedGroupAlgebra C G => a • x) := by
funext a
exact
kerAugClosedQuotTargetCompletedModuleOfSurj_map_smul
C hC hForm psi hpsi hfopen a x
simpa [hEq] using hsourceProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem continuous_zcCompletedGroupAlgebraKernelAugmentationClosedQuotient_source_smul
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi) :
Continuous (fun p : ZCCompletedGroupAlgebra C G ×
KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen =>
p.1 • p.2)Source scalar multiplication on the closed source augmentation quotient is jointly continuous.
Show proof
by
let S :=
zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen
have hquot :
IsOpenQuotientMap
(Prod.map (id : ZCCompletedGroupAlgebra C G → ZCCompletedGroupAlgebra C G)
(fun z : zcCompletedGroupAlgebraStandardAugmentationIdeal C G =>
(Submodule.Quotient.mk (p := S) z :
KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen))) :=
IsOpenQuotientMap.id.prodMap S.isOpenQuotientMap_mkQ
rw [← hquot.continuous_comp_iff]
have hpre :
Continuous (fun p : ZCCompletedGroupAlgebra C G ×
zcCompletedGroupAlgebraStandardAugmentationIdeal C G =>
(⟨p.1 * (p.2 : ZCCompletedGroupAlgebra C G),
(zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left p.1 p.2.2⟩ :
zcCompletedGroupAlgebraStandardAugmentationIdeal C G)) := by
have hmul : Continuous (fun p : ZCCompletedGroupAlgebra C G ×
zcCompletedGroupAlgebraStandardAugmentationIdeal C G =>
p.1 * (p.2 : ZCCompletedGroupAlgebra C G)) :=
continuous_fst.mul (continuous_subtype_val.comp continuous_snd)
exact Continuous.subtype_mk hmul
(fun p => (zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left p.1 p.2.2)
have hmk :
Continuous (fun z : zcCompletedGroupAlgebraStandardAugmentationIdeal C G =>
(Submodule.Quotient.mk (p := S) z :
KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)) :=
continuous_quotient_mk'
simpa [Function.comp_def, Prod.map, S] using hmk.comp hpreProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem continuous_kerAugClosedQuotTargetCompletedModuleOfSurj_smul
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi) :
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)Descended target scalar multiplication on the closed source augmentation quotient is jointly continuous.
Show proof
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
Continuous (fun p : ZCCompletedGroupAlgebra C H ×
KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen =>
p.1 • p.2) := by
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
let q := zcCompletedGroupAlgebraMap C hC psi
have hq : IsOpenQuotientMap q :=
isOpenQuotientMap_zcCompletedGroupAlgebraMap_of_surjective C hC hForm psi hpsi
have hquot :
IsOpenQuotientMap
(Prod.map q
(id :
KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen →
KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)) :=
hq.prodMap IsOpenQuotientMap.id
rw [← hquot.continuous_comp_iff]
have hsource :=
continuous_zcCompletedGroupAlgebraKernelAugmentationClosedQuotient_source_smul
C hC hForm psi hpsi hfopen
have hEq :
(fun p : ZCCompletedGroupAlgebra C G ×
KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen =>
q p.1 • p.2) =
(fun p : ZCCompletedGroupAlgebra C G ×
KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen =>
p.1 • p.2) := by
funext p
exact
kerAugClosedQuotTargetCompletedModuleOfSurj_map_smul
C hC hForm psi hpsi hfopen p.1 p.2
simpa [Function.comp_def, Prod.map, hEq] using hsourceProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem continuousSMul_zcCompletedGroupAlgebraKernelAugmentationClosedQuotient_source
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi) :
ContinuousSMul (ZCCompletedGroupAlgebra C G)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) where
continuous_smulThe closed source augmentation quotient is a topological module for the source completed group algebra.
Show proof
continuous_zcCompletedGroupAlgebraKernelAugmentationClosedQuotient_source_smul
C hC hForm psi hpsi hfopenProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem continuousSMul_kerAugClosedQuotTargetCompletedModuleOfSurj
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi) :
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)The descended target module structure on the closed source augmentation quotient is topological.
Show proof
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
ContinuousSMul (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) := by
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
exact
⟨continuous_kerAugClosedQuotTargetCompletedModuleOfSurj_smul
C hC hForm psi hpsi hfopen⟩Proof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem kerAugClosedQuotTargetCompletedModuleOfSurj_groupLike_smul
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(g : G) (x : KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)Group-like elements act on the closed quotient target module by the induced target action.
Show proof
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
zcGroupLike C H (psi g) • x = zcGroupLike C G g • x := by
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
rw [← zcCompletedGroupAlgebraMap_groupLike (C := C) (hC := hC) psi g]
exact
kerAugClosedQuotTargetCompletedModuleOfSurj_map_smul
C hC hForm psi hpsi hfopen (zcGroupLike C G g) xProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem zcCompletedGASourceBoundaryToKerAugClosedQuot_isTargetCompletedCrossedDiff_of_surj
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi) :
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)The source boundary to the closed source augmentation quotient is a crossed differential for the descended completed target scalars.
Show proof
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
IsCrossedDifferential
(zcCompletedGroupAlgebraScalar C psi.toMonoidHom)
(zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
C hC hForm psi hpsi hfopen) := by
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
intro g h
have hsource :=
zcCompletedGASourceBoundaryToKerAugClosedQuot_isCrossedDiff
C hC hForm psi hpsi hfopen g h
rw [hsource]
congr 1
change zcGroupLike C G g •
zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
C hC hForm psi hpsi hfopen h =
zcGroupLike C H (psi g) •
zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
C hC hForm psi hpsi hfopen h
exact
(kerAugClosedQuotTargetCompletedModuleOfSurj_groupLike_smul
C hC hForm psi hpsi hfopen g
(zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
C hC hForm psi hpsi hfopen h)).symmProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem zcCompletedGroupAlgebraOpenImageStageRingHom_projection_targetLift
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(i : ZCCompletedGroupAlgebraIndex C G)
(a : ZCCompletedGroupAlgebra C H) :
zcCompletedGroupAlgebraOpenImageStageRingHom C hC hForm psi hpsi hfopen i
(zcCompletedGroupAlgebraProjection C G i
(zcCompletedGroupAlgebraTargetLiftOfSurjective C hC hForm psi hpsi a)) =
zcCompletedGroupAlgebraProjection C H
(zcCompletedGroupAlgebraOpenImageTargetIndex C hForm psi hpsi hfopen i) aProjecting a chosen completed source lift of a target coefficient to a source stage and then passing to the open-image stage recovers the corresponding target finite-stage projection.
Show proof
by
let b := zcCompletedGroupAlgebraTargetLiftOfSurjective C hC hForm psi hpsi a
have hsource :
(i.1,
completedGroupAlgebraComapIndexInClass (G := G) (H := H) C hC psi
(zcCompletedGroupAlgebraOpenImageIndexInClass C hForm psi hpsi hfopen i)) ≤ i :=
⟨le_rfl, zcCompletedGroupAlgebraOpenImage_comapIndex_le C hC hForm psi hpsi hfopen i⟩
have hstage :
zcCompletedGroupAlgebraOpenImageStageRingHom C hC hForm psi hpsi hfopen i
(zcCompletedGroupAlgebraProjection C G i b) =
zcCompletedGroupAlgebraMapStage C hC psi
(zcCompletedGroupAlgebraOpenImageTargetIndex C hForm psi hpsi hfopen i)
(zcCompletedGroupAlgebraTransition C G hsource
(zcCompletedGroupAlgebraProjection C G i b)) := by
exact congrFun
(congrArg DFunLike.coe
(zcCompletedGroupAlgebraOpenImageQuotientMap_stage_eq
C hC hForm psi hpsi hfopen i))
(zcCompletedGroupAlgebraProjection C G i b)
rw [hstage, zcCompletedGroupAlgebraTransition_projection]
change
zcCompletedGroupAlgebraMapStage C hC psi
(zcCompletedGroupAlgebraOpenImageTargetIndex C hForm psi hpsi hfopen i)
(zcCompletedGroupAlgebraProjection C G
((zcCompletedGroupAlgebraOpenImageTargetIndex C hForm psi hpsi hfopen i).1,
completedGroupAlgebraComapIndexInClass (G := G) (H := H) C hC psi
(zcCompletedGroupAlgebraOpenImageTargetIndex C hForm psi hpsi hfopen i).2) b) =
zcCompletedGroupAlgebraProjection C H
(zcCompletedGroupAlgebraOpenImageTargetIndex C hForm psi hpsi hfopen i) a
have hprojmap :=
zcCompletedGroupAlgebraProjection_map
(C := C) (hC := hC) (H := G) (K := H) (φ := psi)
(i := zcCompletedGroupAlgebraOpenImageTargetIndex C hForm psi hpsi hfopen i)
(x := b)
have hmap : zcCompletedGroupAlgebraMap C hC psi b = a := by
dsimp [b]
exact zcCompletedGroupAlgebraMap_targetLiftOfSurjective C hC hForm psi hpsi a
have hmain :
zcCompletedGroupAlgebraProjection C H
(zcCompletedGroupAlgebraOpenImageTargetIndex C hForm psi hpsi hfopen i)
(zcCompletedGroupAlgebraMap C hC psi b) =
zcCompletedGroupAlgebraProjection C H
(zcCompletedGroupAlgebraOpenImageTargetIndex C hForm psi hpsi hfopen i) a :=
congrArg
(zcCompletedGroupAlgebraProjection C H
(zcCompletedGroupAlgebraOpenImageTargetIndex C hForm psi hpsi hfopen i))
hmap
exact hprojmap.symm.trans hmainProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem kerAugIdealClosedQuotStageProj_liftLinear_eq_boundaryLift_preStageMap
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(i : ZCCompletedGroupAlgebraIndex C G)
(x : CrossedDifferentialPreModule (ZCCompletedGroupAlgebra C H) G) :
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)The i-th closed augmentation quotient coordinate of the completed source-boundary lift factors through the corresponding open-image finite pre-stage.
Show proof
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
kernelAugmentationIdealClosedQuotientStageProjection
C hC hForm psi hpsi hfopen i
(crossedDifferentialModuleLiftLinear
(R := ZCCompletedGroupAlgebra C H)
(zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
C hC hForm psi hpsi hfopen) x) =
kernelAugmentationIdealClosedStageQuotientBoundaryLift
C hC hForm psi hpsi hfopen i
(zcCompletedDifferentialModulePreStageMap C psi.toMonoidHom
(zcCompletedDifferentialModuleOpenImageIndex C hC hForm psi hpsi hfopen i) x) := by
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
let j := zcCompletedDifferentialModuleOpenImageIndex C hC hForm psi hpsi hfopen i
letI : Module (zcCompletedDifferentialModuleStageRing C psi.toMonoidHom j)
(KernelAugmentationIdealClosedStageQuotient C hC hForm psi hpsi hfopen i) :=
kernelAugmentationIdealClosedStageQuotientTargetStageModule
C hC hForm psi hpsi hfopen i
letI : Module
(ZCCompletedGroupAlgebraStage C H
(zcCompletedGroupAlgebraOpenImageTargetIndex C hForm psi hpsi hfopen i))
(KernelAugmentationIdealClosedStageQuotient C hC hForm psi hpsi hfopen i) :=
kernelAugmentationIdealClosedStageQuotientTargetStageModule
C hC hForm psi hpsi hfopen i
refine Finsupp.induction_linear x ?zero ?add ?single
· simp only [crossedDifferentialModuleLiftLinear, map_zero, ContinuousMonoidHom.coe_toMonoidHom,
Lean.Elab.WF.paramLet, kernelAugmentationIdealClosedStageQuotientBoundaryLift,
zcCompletedDifferentialModulePreStageMap]
· intro x y hx hy
simp only [map_add, hx, ContinuousMonoidHom.coe_toMonoidHom, Lean.Elab.WF.paramLet, hy]
· intro g a
rw [crossedDifferentialModuleLiftLinear_single,
zcCompletedDifferentialModulePreStageMap_single,
kernelAugmentationIdealClosedStageQuotientBoundaryLift_single]
let b := zcCompletedGroupAlgebraTargetLiftOfSurjective C hC hForm psi hpsi a
have htarget :
a • zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
C hC hForm psi hpsi hfopen g =
b • zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
C hC hForm psi hpsi hfopen g := by
rfl
rw [htarget]
change
kernelAugmentationIdealClosedQuotientStageProjection
C hC hForm psi hpsi hfopen i
(b • zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
C hC hForm psi hpsi hfopen g) =
zcCompletedGroupAlgebraProjection C H
(zcCompletedGroupAlgebraOpenImageTargetIndex C hForm psi hpsi hfopen i) a •
kernelAugmentationIdealClosedStageQuotientBoundary
C hC hForm psi hpsi hfopen i
(zcCompletedDifferentialModuleStageSourceProj C psi.toMonoidHom j g)
rw [map_smulₛₗ]
rw [← zcCompletedGroupAlgebraOpenImageStageRingHom_projection_targetLift
C hC hForm psi hpsi hfopen i a]
rw [kernelAugmentationIdealClosedStageQuotientTargetStageModule_map_smul
(C := C) (hC := hC) (hForm := hForm) (psi := psi)
(hpsi := hpsi) (hfopen := hfopen) (i := i)
(a := zcCompletedGroupAlgebraProjection C G i b)
(x := kernelAugmentationIdealClosedStageQuotientBoundary
C hC hForm psi hpsi hfopen i
(zcCompletedDifferentialModuleStageSourceProj C psi.toMonoidHom j g))]
have hstage_boundary :
kernelAugmentationIdealClosedQuotientStageProjection
C hC hForm psi hpsi hfopen i
(zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
C hC hForm psi hpsi hfopen g) =
kernelAugmentationIdealClosedStageQuotientBoundary
C hC hForm psi hpsi hfopen i
(zcCompletedDifferentialModuleStageSourceProj C psi.toMonoidHom j g) := by
rw [zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient]
let s : zcCompletedGroupAlgebraStandardAugmentationIdeal C G :=
⟨zcCompletedGroupAlgebraBoundary C (MonoidHom.id G) g,
zcCompletedGroupAlgebraBoundary_mem_standardAugmentationIdeal
C G (MonoidHom.id G) g⟩
change
kernelAugmentationIdealClosedQuotientStageProjection
C hC hForm psi hpsi hfopen i
(Submodule.Quotient.mk
(p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen) s) =
kernelAugmentationIdealClosedStageQuotientBoundary
C hC hForm psi hpsi hfopen i
(zcCompletedDifferentialModuleStageSourceProj C psi.toMonoidHom j g)
let T :=
zcCompletedGroupAlgebraOpenImageKernelAugmentationIdealMulStageStandard
C hC hForm psi hpsi hfopen i
calc
kernelAugmentationIdealClosedQuotientStageProjection
C hC hForm psi hpsi hfopen i
(Submodule.Quotient.mk
(p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen) s) =
(Submodule.Quotient.mk
(p := T)
(zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i s) :
KernelAugmentationIdealClosedStageQuotient C hC hForm psi hpsi hfopen i) := by
exact
kernelAugmentationIdealClosedQuotientStageProjection_mk
C hC hForm psi hpsi hfopen i s
_ =
kernelAugmentationIdealClosedStageQuotientBoundary
C hC hForm psi hpsi hfopen i
(zcCompletedDifferentialModuleStageSourceProj C psi.toMonoidHom j g) := by
apply congrArg (fun y : zcCompletedGroupAlgebraStageAugmentationIdeal C G i =>
(Submodule.Quotient.mk (p := T) y :
KernelAugmentationIdealClosedStageQuotient C hC hForm psi hpsi hfopen i))
apply Subtype.ext
simp only [zcCompletedGroupAlgebraStandardAugmentationIdealProjection, zcCompletedGroupAlgebraBoundary,
MonoidHom.id_apply, zcCompletedGroupAlgebraAugmentationIdealProjection_val, zcCompletedGroupAlgebraProjection_sub,
zcCompletedGroupAlgebraProjection_groupLike, MonoidAlgebra.of_apply, zcCompletedGroupAlgebraProjection_one,
zcCompletedGroupAlgebraStageAugmentationGeneratorSubtype, zcCompletedGroupAlgebraStageAugmentationGenerator,
ContinuousMonoidHom.coe_toMonoidHom, zcCompletedDifferentialModuleOpenImageIndex,
zcCompletedDifferentialModuleStageSourceProj, QuotientGroup.mk'_apply, s, j]
rw [hstage_boundary]
rflProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem continuous_kernelAugmentationIdealClosedQuotientStageProjection_liftLinear
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(i : ZCCompletedGroupAlgebraIndex C G) :
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)Each finite closed-augmentation coordinate of the pre-quotient source-boundary lift is continuous for the finite-stage pre-module topology.
Show proof
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
@Continuous
(CrossedDifferentialPreModule (ZCCompletedGroupAlgebra C H) G)
(KernelAugmentationIdealClosedStageQuotient C hC hForm psi hpsi hfopen i)
(zcCompletedDifferentialPreModuleNaturalTopology C psi.toMonoidHom)
inferInstance
(fun x =>
kernelAugmentationIdealClosedQuotientStageProjection
C hC hForm psi hpsi hfopen i
(crossedDifferentialModuleLiftLinear
(R := ZCCompletedGroupAlgebra C H)
(zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
C hC hForm psi hpsi hfopen) x)) := by
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
let j := zcCompletedDifferentialModuleOpenImageIndex C hC hForm psi hpsi hfopen i
letI : Module (zcCompletedDifferentialModuleStageRing C psi.toMonoidHom j)
(KernelAugmentationIdealClosedStageQuotient C hC hForm psi hpsi hfopen i) :=
kernelAugmentationIdealClosedStageQuotientTargetStageModule
C hC hForm psi hpsi hfopen i
letI : TopologicalSpace
(CrossedDifferentialPreModule (ZCCompletedGroupAlgebra C H) G) :=
zcCompletedDifferentialPreModuleNaturalTopology C psi.toMonoidHom
letI : TopologicalSpace
(CrossedDifferentialPreModule
(zcCompletedDifferentialModuleStageRing C psi.toMonoidHom j)
(zcCompletedDifferentialModuleStageSource C psi.toMonoidHom j)) :=
⊥
letI : DiscreteTopology
(CrossedDifferentialPreModule
(zcCompletedDifferentialModuleStageRing C psi.toMonoidHom j)
(zcCompletedDifferentialModuleStageSource C psi.toMonoidHom j)) :=
⟨rfl⟩
have hpre :
@Continuous
(CrossedDifferentialPreModule (ZCCompletedGroupAlgebra C H) G)
(CrossedDifferentialPreModule
(zcCompletedDifferentialModuleStageRing C psi.toMonoidHom j)
(zcCompletedDifferentialModuleStageSource C psi.toMonoidHom j))
(zcCompletedDifferentialPreModuleNaturalTopology C psi.toMonoidHom)
inferInstance
(zcCompletedDifferentialModulePreStageMap C psi.toMonoidHom j) :=
continuous_zcCompletedDifferentialModulePreStageMap_naturalTopology
C psi.toMonoidHom j
have hfinite :
Continuous
(kernelAugmentationIdealClosedStageQuotientBoundaryLift
C hC hForm psi hpsi hfopen i) :=
continuous_of_discreteTopology
have hfactor :
(fun x : CrossedDifferentialPreModule (ZCCompletedGroupAlgebra C H) G =>
kernelAugmentationIdealClosedQuotientStageProjection
C hC hForm psi hpsi hfopen i
(crossedDifferentialModuleLiftLinear
(R := ZCCompletedGroupAlgebra C H)
(zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
C hC hForm psi hpsi hfopen) x)) =
fun x =>
kernelAugmentationIdealClosedStageQuotientBoundaryLift
C hC hForm psi hpsi hfopen i
(zcCompletedDifferentialModulePreStageMap C psi.toMonoidHom j x) := by
funext x
exact
kerAugIdealClosedQuotStageProj_liftLinear_eq_boundaryLift_preStageMap
C hC hForm psi hpsi hfopen i x
rw [hfactor]
exact hfinite.comp hpreProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem continuous_crossedDiffModuleLiftLinear_sourceBoundaryToKerAugClosedQuot
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi) :
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)The pre-quotient source-boundary lift to the closed source augmentation quotient is continuous for the finite-stage pre-module topology.
Show proof
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
@Continuous
(CrossedDifferentialPreModule (ZCCompletedGroupAlgebra C H) G)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)
(zcCompletedDifferentialPreModuleNaturalTopology C psi.toMonoidHom)
inferInstance
(crossedDifferentialModuleLiftLinear
(R := ZCCompletedGroupAlgebra C H)
(zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
C hC hForm psi hpsi hfopen)) := by
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
letI : TopologicalSpace
(CrossedDifferentialPreModule (ZCCompletedGroupAlgebra C H) G) :=
zcCompletedDifferentialPreModuleNaturalTopology C psi.toMonoidHom
rw [kernelAugmentationIdealClosedQuotient_topology_eq_induced_stageProjProduct
C hC hForm psi hpsi hfopen]
rw [continuous_induced_rng]
exact continuous_pi fun i =>
continuous_kernelAugmentationIdealClosedQuotientStageProjection_liftLinear
C hC hForm psi hpsi hfopen iProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem zcSeparatedUniversalDifferential_isSourceCompletedCrossedDifferential
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(psi : ContinuousMonoidHom G H) :
letI : Module (ZCCompletedGroupAlgebra C G)
(ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom)The separated universal differential is also a crossed differential for source completed group-algebra scalars after restricting scalars along \(\mathbb{Z}_C\llbracket G\rrbracket \to \mathbb{Z}_C\llbracket H\rrbracket\).
Show proof
Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
IsCrossedDifferential
(zcCompletedGroupAlgebraScalar C (MonoidHom.id G))
(zcSeparatedUniversalDifferential C psi.toMonoidHom) := by
letI : Module (ZCCompletedGroupAlgebra C G)
(ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
intro g h
rw [zcSeparatedUniversalDifferential_mul]
congr 1
change zcGroupLike C H (psi g) •
zcSeparatedUniversalDifferential C psi.toMonoidHom h =
zcCompletedGroupAlgebraMap C hC psi (zcGroupLike C G g) •
zcSeparatedUniversalDifferential C psi.toMonoidHom h
rw [zcCompletedGroupAlgebraMap_groupLike]Proof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□def zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(psi : ContinuousMonoidHom G H) :
letI : Module (ZCCompletedGroupAlgebra C G)
(ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
ZCCompletedDifferentialModule C (MonoidHom.id G) →ₗ[ZCCompletedGroupAlgebra C G]
ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom := by
letI : Module (ZCCompletedGroupAlgebra C G)
(ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
exact
zcCompletedDifferentialModuleLift
(A := ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom)
C (MonoidHom.id G)
(zcSeparatedUniversalDifferential C psi.toMonoidHom)
(zcSeparatedUniversalDifferential_isSourceCompletedCrossedDifferential C hC psi)
@[simp]The source-identity completed differential module maps to the separated module for \(\psi\) by \(dg \mapsto\) \(d_{\mathrm{sep}}\) g, with source scalars restricted through \(\mathbb{Z}_C\llbracket G\rrbracket\) \(\to\) \(\mathbb{Z}_C\llbracket H\rrbracket\).
theorem zcDiffModuleIdToZCSepDiffModule_universal
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(psi : ContinuousMonoidHom G H) (g : G) :
zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule C hC psi
(zcUniversalDifferential C (MonoidHom.id G) g) =
zcSeparatedUniversalDifferential C psi.toMonoidHom gThe comparison map from the identity differential module to the separated completed module sends the universal differential to the separated universal differential.
Show proof
by
letI : Module (ZCCompletedGroupAlgebra C G)
(ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
exact
zcCompletedDifferentialModuleLift_universal
(A := ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom)
C (MonoidHom.id G)
(zcSeparatedUniversalDifferential C psi.toMonoidHom)
(zcSeparatedUniversalDifferential_isSourceCompletedCrossedDifferential C hC psi) gProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem zcCompletedDifferentialModuleIdentitySourceStageRingHom_transition_mapStage
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(psi : ContinuousMonoidHom G H)
(i : ZCCompletedDifferentialModuleIndex C psi.toMonoidHom) :
let sourceIndex : ZCCompletedGroupAlgebraIndex C GShow proof
(i.target.1, completedGroupAlgebraComapIndexInClass
(G := G) (H := H) C hC psi i.target.2)
let idIndex := zcCompletedDifferentialModuleIdentitySourceIndex C psi.toMonoidHom i
let hle : sourceIndex ≤ idIndex.target := by
constructor
exact le_rfl
exact i.compatible
∀ x : ZCCompletedGroupAlgebraStage C G idIndex.target,
zcCompletedGroupAlgebraMapStage C hC psi i.target
(zcCompletedGroupAlgebraTransition C G hle x) =
zcCompletedDifferentialModuleIdentitySourceStageRingHom C psi.toMonoidHom i x := by
intro sourceIndex idIndex hle x
refine MonoidAlgebra.induction_on
(p := fun x : ZCCompletedGroupAlgebraStage C G idIndex.target =>
zcCompletedGroupAlgebraMapStage C hC psi i.target
(zcCompletedGroupAlgebraTransition C G hle x) =
zcCompletedDifferentialModuleIdentitySourceStageRingHom C psi.toMonoidHom i x)
x ?single ?add ?smul
· intro q
refine QuotientGroup.induction_on q ?_
intro g
rw [zcCompletedGroupAlgebraTransition_of]
dsimp [sourceIndex, idIndex]
rw [zcCompletedGroupAlgebraMapStage_single]
change MonoidAlgebra.single
((completedGroupAlgebraComapQuotientMapInClass (G := G) (H := H) C hC psi i.target.2)
((OpenNormalSubgroupInClass.map (C := C) (G := G)
(U := OrderDual.ofDual (completedGroupAlgebraComapIndexInClass
(G := G) (H := H) C hC psi i.target.2))
(V := i.source) i.compatible)
(QuotientGroup.mk' (i.source.1 : Subgroup G) g))) 1 =
MonoidAlgebra.mapDomain (zcCompletedDifferentialModuleStagePsi C psi.toMonoidHom i)
(Finsupp.single (QuotientGroup.mk' (i.source.1 : Subgroup G) g) 1)
rw [MonoidAlgebra.mapDomain_single]
dsimp [OpenNormalSubgroupInClass.map]
change MonoidAlgebra.single
((completedGroupAlgebraComapQuotientMapInClass (G := G) (H := H) C hC psi i.target.2)
(QuotientGroup.mk'
((((OrderDual.ofDual (completedGroupAlgebraComapIndexInClass
(G := G) (H := H) C hC psi i.target.2)).1 :
OpenNormalSubgroup G) : Subgroup G)) g)) 1 =
MonoidAlgebra.single
((zcCompletedDifferentialModuleStagePsi C psi.toMonoidHom i)
(QuotientGroup.mk' (i.source.1 : Subgroup G) g)) 1
rw [completedGroupAlgebraComapQuotientMapInClass_mk]
change MonoidAlgebra.single (QuotientGroup.mk'
((((OrderDual.ofDual i.target.2).1 : OpenNormalSubgroup H) : Subgroup H))
(psi g)) 1 =
MonoidAlgebra.single
((QuotientGroup.map (i.source.1 : Subgroup G)
((((OrderDual.ofDual i.target.2).1 : OpenNormalSubgroup H) : Subgroup H))
psi.toMonoidHom i.compatible)
(QuotientGroup.mk' (i.source.1 : Subgroup G) g)) 1
rw [QuotientGroup.map_mk']
rfl
· intro x y hx hy
simp only [ContinuousMonoidHom.coe_toMonoidHom, map_add, hx,
zcCompletedDifferentialModuleIdentitySourceIndex_target_fst, hy]
· intro r x hx
rcases ZMod.intCast_surjective r with ⟨t, rfl⟩
rw [Algebra.smul_def, RingHom.map_mul, RingHom.map_mul, hx]
simp only [ContinuousMonoidHom.coe_toMonoidHom, zcCompletedGroupAlgebraMapStage,
zcCompletedGroupAlgebraTransition, modNCompletedGroupAlgebraStageCoeffMapInClass, modNCompletedGroupRingCoeffMap,
AlgHom.toRingHom_eq_coe, map_intCast, zcCompletedDifferentialModuleIdentitySourceIndex_target_fst,
zcCompletedDifferentialModuleIdentitySourceStageRingHom, map_mul]Proof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem zcCompletedDifferentialModuleIdentitySourceStageRingHom_projection_map
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(psi : ContinuousMonoidHom G H)
(i : ZCCompletedDifferentialModuleIndex C psi.toMonoidHom)
(a : ZCCompletedGroupAlgebra C G) :
zcCompletedDifferentialModuleIdentitySourceStageRingHom C psi.toMonoidHom i
(zcCompletedGroupAlgebraProjection C G
(zcCompletedDifferentialModuleIdentitySourceIndex C psi.toMonoidHom i).target a) =
zcCompletedGroupAlgebraProjection C H i.target
(zcCompletedGroupAlgebraMap C hC psi a)Completed source coefficients viewed at the identity-source stage agree with target finite projections after applying the completed group-algebra map.
Show proof
by
let sourceIndex : ZCCompletedGroupAlgebraIndex C G :=
(i.target.1, completedGroupAlgebraComapIndexInClass
(G := G) (H := H) C hC psi i.target.2)
let idIndex := zcCompletedDifferentialModuleIdentitySourceIndex C psi.toMonoidHom i
have hle : sourceIndex ≤ idIndex.target := by
constructor
exact le_rfl
exact i.compatible
rw [zcCompletedGroupAlgebraProjection_map]
rw [← zcCompletedDifferentialModuleIdentitySourceStageRingHom_transition_mapStage
C hC psi i (zcCompletedGroupAlgebraProjection C G idIndex.target a)]
rw [zcCompletedGroupAlgebraTransition_projection]Proof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem zcDiffModuleIdToZCSepDiffModule_stageProj
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(psi : ContinuousMonoidHom G H)
(i : ZCCompletedDifferentialModuleIndex C psi.toMonoidHom)
(x : ZCCompletedDifferentialModule C (MonoidHom.id G)) :
zcSeparatedCompletedDifferentialModuleStageProjectionAdd C psi.toMonoidHom i
(zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule C hC psi x) =
zcCompletedDifferentialModuleIdentitySourceStageToStage C psi.toMonoidHom i
(zcCompletedDifferentialModuleStageProjection C (MonoidHom.id G)
(zcCompletedDifferentialModuleIdentitySourceIndex C psi.toMonoidHom i) x)Finite-stage projections of the identity-source lift to the separated \(\psi\)-module are computed by first projecting to the matching source-identity finite stage.
Show proof
by
letI : Module (ZCCompletedGroupAlgebra C G)
(ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
letI : Module (ZCCompletedGroupAlgebra C G)
(ZCCompletedDifferentialModuleStage C psi.toMonoidHom i) :=
Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
let idIndex := zcCompletedDifferentialModuleIdentitySourceIndex C psi.toMonoidHom i
letI : Module
(zcCompletedDifferentialModuleStageRing C (MonoidHom.id G) idIndex)
(ZCCompletedDifferentialModuleStage C psi.toMonoidHom i) :=
Module.compHom _ (zcCompletedDifferentialModuleIdentitySourceStageRingHom C psi.toMonoidHom i)
let L :
ZCCompletedDifferentialModule C (MonoidHom.id G) →ₗ[ZCCompletedGroupAlgebra C G]
ZCCompletedDifferentialModuleStage C psi.toMonoidHom i :=
{ toFun := fun x =>
zcSeparatedCompletedDifferentialModuleStageProjectionAdd C psi.toMonoidHom i
(zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule C hC psi x)
map_add' := by
intro x y
rw [map_add, map_add]
map_smul' := by
intro a x
rw [map_smul]
change
zcSeparatedCompletedDifferentialModuleStageProjectionAdd C psi.toMonoidHom i
(zcCompletedGroupAlgebraMap C hC psi a •
zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule
C hC psi x) =
zcCompletedGroupAlgebraMap C hC psi a •
zcSeparatedCompletedDifferentialModuleStageProjectionAdd C psi.toMonoidHom i
(zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule
C hC psi x)
rw [map_smul] }
let R :
ZCCompletedDifferentialModule C (MonoidHom.id G) →ₗ[ZCCompletedGroupAlgebra C G]
ZCCompletedDifferentialModuleStage C psi.toMonoidHom i :=
{ toFun := fun x =>
zcCompletedDifferentialModuleIdentitySourceStageToStage C psi.toMonoidHom i
(zcCompletedDifferentialModuleStageProjection C (MonoidHom.id G)
(zcCompletedDifferentialModuleIdentitySourceIndex C psi.toMonoidHom i) x)
map_add' := by
intro x y
rw [map_add, map_add]
map_smul' := by
intro a x
rw [map_smul]
change
zcCompletedDifferentialModuleIdentitySourceStageToStage C psi.toMonoidHom i
(zcCompletedGroupAlgebraProjection C G
idIndex.target a •
zcCompletedDifferentialModuleStageProjection C (MonoidHom.id G)
idIndex x) =
zcCompletedGroupAlgebraProjection C H i.target
(zcCompletedGroupAlgebraMap C hC psi a) •
zcCompletedDifferentialModuleIdentitySourceStageToStage C psi.toMonoidHom i
(zcCompletedDifferentialModuleStageProjection C (MonoidHom.id G)
idIndex x)
rw [map_smul]
change
zcCompletedDifferentialModuleIdentitySourceStageRingHom C psi.toMonoidHom i
(zcCompletedGroupAlgebraProjection C G idIndex.target a) •
zcCompletedDifferentialModuleIdentitySourceStageToStage C psi.toMonoidHom i
(zcCompletedDifferentialModuleStageProjection C (MonoidHom.id G)
idIndex x) =
zcCompletedGroupAlgebraProjection C H i.target
(zcCompletedGroupAlgebraMap C hC psi a) •
zcCompletedDifferentialModuleIdentitySourceStageToStage C psi.toMonoidHom i
(zcCompletedDifferentialModuleStageProjection C (MonoidHom.id G)
idIndex x)
rw [zcCompletedDifferentialModuleIdentitySourceStageRingHom_projection_map C hC psi i a] }
have hLR : L = R := by
apply zcCompletedDifferentialModuleHom_ext C (MonoidHom.id G)
intro g
change
zcSeparatedCompletedDifferentialModuleStageProjectionAdd C psi.toMonoidHom i
(zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule C hC psi
(zcUniversalDifferential C (MonoidHom.id G) g)) =
zcCompletedDifferentialModuleIdentitySourceStageToStage C psi.toMonoidHom i
(zcCompletedDifferentialModuleStageProjection C (MonoidHom.id G)
idIndex (zcUniversalDifferential C (MonoidHom.id G) g))
rw [zcDiffModuleIdToZCSepDiffModule_universal,
zcSeparatedCompletedDifferentialModuleStageProjectionAdd_universal]
calc
zcCompletedDifferentialModuleStageDifferential C psi.toMonoidHom i g =
zcCompletedDifferentialModuleStageProjection C psi.toMonoidHom i
(zcUniversalDifferential C psi.toMonoidHom g) := by
rw [zcCompletedDifferentialModuleStageProjection_universal]
_ =
zcCompletedDifferentialModuleIdentitySourceStageToStage C psi.toMonoidHom i
(zcCompletedDifferentialModuleStageProjection C (MonoidHom.id G)
idIndex (zcUniversalDifferential C (MonoidHom.id G) g)) := by
exact
(zcDiffModuleIdentitySourceStageToStage_stageProj_universal
C psi.toMonoidHom i g).symm
exact LinearMap.congr_fun hLR xProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem zcCompletedDifferentialModuleStageBoundary_identitySourceStageToStage
(ψ : G →* H)
(i : ZCCompletedDifferentialModuleIndex C ψ)
(x : ZCCompletedDifferentialModuleStage C (MonoidHom.id G)
(zcCompletedDifferentialModuleIdentitySourceIndex C ψ i)) :
zcCompletedDifferentialModuleStageBoundary C ψ i
(zcCompletedDifferentialModuleIdentitySourceStageToStage C ψ i x) =
zcCompletedDifferentialModuleIdentitySourceStageRingHom C ψ i
(zcCompletedDifferentialModuleStageBoundary C (MonoidHom.id G)
(zcCompletedDifferentialModuleIdentitySourceIndex C ψ i) x)The finite comparison from the source-identity stage to the \(\psi\)-stage commutes with the finite Fox boundary.
Show proof
by
let j := zcCompletedDifferentialModuleIdentitySourceIndex C ψ i
letI : Module (zcCompletedDifferentialModuleStageRing C (MonoidHom.id G) j)
(zcCompletedDifferentialModuleStageRing C ψ i) :=
Module.compHom _ (zcCompletedDifferentialModuleIdentitySourceStageRingHom C ψ i)
letI : Module (zcCompletedDifferentialModuleStageRing C (MonoidHom.id G) j)
(ZCCompletedDifferentialModuleStage C ψ i) :=
Module.compHom _ (zcCompletedDifferentialModuleIdentitySourceStageRingHom C ψ i)
let ringMapLinear :
zcCompletedDifferentialModuleStageRing C (MonoidHom.id G) j →ₗ[
zcCompletedDifferentialModuleStageRing C (MonoidHom.id G) j]
zcCompletedDifferentialModuleStageRing C ψ i := {
toFun := zcCompletedDifferentialModuleIdentitySourceStageRingHom C ψ i
map_add' := by
intro a b
exact map_add (zcCompletedDifferentialModuleIdentitySourceStageRingHom C ψ i) a b
map_smul' := by
intro a b
change
zcCompletedDifferentialModuleIdentitySourceStageRingHom C ψ i (a * b) =
zcCompletedDifferentialModuleIdentitySourceStageRingHom C ψ i a *
zcCompletedDifferentialModuleIdentitySourceStageRingHom C ψ i b
exact map_mul (zcCompletedDifferentialModuleIdentitySourceStageRingHom C ψ i) a b }
let L :
ZCCompletedDifferentialModuleStage C (MonoidHom.id G) j →ₗ[
zcCompletedDifferentialModuleStageRing C (MonoidHom.id G) j]
zcCompletedDifferentialModuleStageRing C ψ i := {
toFun := fun x =>
zcCompletedDifferentialModuleStageBoundary C ψ i
(zcCompletedDifferentialModuleIdentitySourceStageToStage C ψ i x)
map_add' := by
intro x y
rw [map_add, map_add]
map_smul' := by
intro a x
rw [map_smul]
change
zcCompletedDifferentialModuleStageBoundary C ψ i
(zcCompletedDifferentialModuleIdentitySourceStageRingHom C ψ i a •
zcCompletedDifferentialModuleIdentitySourceStageToStage C ψ i x) =
zcCompletedDifferentialModuleIdentitySourceStageRingHom C ψ i a •
zcCompletedDifferentialModuleStageBoundary C ψ i
(zcCompletedDifferentialModuleIdentitySourceStageToStage C ψ i x)
rw [map_smul] }
let R :
ZCCompletedDifferentialModuleStage C (MonoidHom.id G) j →ₗ[
zcCompletedDifferentialModuleStageRing C (MonoidHom.id G) j]
zcCompletedDifferentialModuleStageRing C ψ i :=
ringMapLinear.comp
(zcCompletedDifferentialModuleStageBoundary C (MonoidHom.id G) j)
have hLR : L = R := by
apply crossedDifferentialModuleHom_ext
(coeff := zcCompletedDifferentialModuleStageScalar C (MonoidHom.id G) j)
intro q
change
zcCompletedDifferentialModuleStageBoundary C ψ i
(zcCompletedDifferentialModuleIdentitySourceStageToStage C ψ i
(universalCrossedDifferential
(zcCompletedDifferentialModuleStageScalar C (MonoidHom.id G) j) q)) =
zcCompletedDifferentialModuleIdentitySourceStageRingHom C ψ i
(zcCompletedDifferentialModuleStageBoundary C (MonoidHom.id G) j
(universalCrossedDifferential
(zcCompletedDifferentialModuleStageScalar C (MonoidHom.id G) j) q))
rw [zcCompletedDifferentialModuleIdentitySourceStageToStage_universal]
have hboundaryTarget :
zcCompletedDifferentialModuleStageBoundary C ψ i
(universalCrossedDifferential
(zcCompletedDifferentialModuleStageScalar C ψ i) q) =
zcCompletedDifferentialModuleStageScalar C ψ i q - 1 := by
rw [zcCompletedDifferentialModuleStageBoundary,
crossedDifferentialModuleLift_universal]
have hboundarySource :
zcCompletedDifferentialModuleStageBoundary C (MonoidHom.id G) j
(universalCrossedDifferential
(zcCompletedDifferentialModuleStageScalar C (MonoidHom.id G) j) q) =
zcCompletedDifferentialModuleStageScalar C (MonoidHom.id G) j q - 1 := by
rw [zcCompletedDifferentialModuleStageBoundary,
crossedDifferentialModuleLift_universal]
rw [hboundaryTarget, hboundarySource]
rw [map_sub, map_one,
zcCompletedDifferentialModuleIdentitySourceStageRingHom_stageScalar]
exact LinearMap.congr_fun hLR xProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem zcCompletedDifferentialModuleIdentitySourceStageBoundary_stageProj
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(psi : ContinuousMonoidHom G H)
(i : ZCCompletedDifferentialModuleIndex C psi.toMonoidHom)
(x : ZCCompletedDifferentialModule C (MonoidHom.id G)) :
let jApplying the finite identity boundary after the source-identity finite projection recovers the finite projection of the standard augmentation-valued completed Fox tail.
Show proof
zcCompletedDifferentialModuleIdentitySourceIndex C psi.toMonoidHom i
zcCompletedDifferentialModuleStageBoundary C (MonoidHom.id G) j
(zcCompletedDifferentialModuleStageProjection C (MonoidHom.id G) j x) =
((zcCompletedGroupAlgebraStandardAugmentationIdealProjection C j.target
(zcToStdAugIdeal C G (MonoidHom.id G) x) :
zcCompletedGroupAlgebraStageAugmentationIdeal C G j.target) :
ZCCompletedGroupAlgebraStage C G j.target) := by
intro j
have hcomp := congrArg (fun f => f x)
(zcDiffModuleStageBoundaryCompletedLinearMap_comp_stageProj
C (MonoidHom.id G) j)
simpa [LinearMap.comp_apply,
zcToStdAugIdeal_val] using hcompProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem zcDiffModuleIdToZCSepDiffModule_eq_zero_of_zcToStandard_eq_zero
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(psi : ContinuousMonoidHom G H)
(x : ZCCompletedDifferentialModule C (MonoidHom.id G))
(hx :
zcToStdAugIdeal C G (MonoidHom.id G) x = 0) :
zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule C hC psi x = 0The identity-source lift to the separated \(\psi\)-module kills the kernel of the standard augmentation-valued completed Fox tail.
Show proof
by
have hxFox : zcToCompletedGroupAlgebra C (MonoidHom.id G) x = 0 := by
have hxval :=
congrArg
(fun y : zcCompletedGroupAlgebraStandardAugmentationIdeal C G =>
(y : ZCCompletedGroupAlgebra C G)) hx
simpa [zcToStdAugIdeal_val] using hxval
apply zcSeparatedCompletedDifferentialModuleStageProjectionsSeparate C psi.toMonoidHom
intro i
rw [zcDiffModuleIdToZCSepDiffModule_stageProj]
have hsource :
zcCompletedDifferentialModuleStageProjection C (MonoidHom.id G)
(zcCompletedDifferentialModuleIdentitySourceIndex C psi.toMonoidHom i) x = 0 :=
zcDiffModuleIdentitySourceStageProj_eq_zero_of_zcTo_eq_zero
C psi.toMonoidHom i x hxFox
rw [hsource, map_zero]Proof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem continuous_zcToStdAugIdeal_naturalTopology
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(psi : ContinuousMonoidHom G H) :
@Continuous
(ZCCompletedDifferentialModule C psi.toMonoidHom)
(zcCompletedGroupAlgebraStandardAugmentationIdeal C H)
(zcCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom)
inferInstance
(zcToStdAugIdeal C H psi.toMonoidHom)The standard-augmentation-valued completed Fox tail is continuous for the finite-stage natural topology on the completed differential module.
Show proof
by
letI : TopologicalSpace (ZCCompletedDifferentialModule C psi.toMonoidHom) :=
zcCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom
have hval :
@Continuous
(ZCCompletedDifferentialModule C psi.toMonoidHom)
(ZCCompletedGroupAlgebra C H)
(zcCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom)
inferInstance
(fun x =>
(zcToStdAugIdeal
C H psi.toMonoidHom x : ZCCompletedGroupAlgebra C H)) := by
simpa [zcToStdAugIdeal_val] using
(continuous_zcToCompletedGroupAlgebra_naturalTopology C hC psi)
exact Continuous.subtype_mk hval
(fun x => (zcToStdAugIdeal
C H psi.toMonoidHom x).2)Proof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem zcSeparatedCompletedDifferentialModule_source_kernel_groupLike_smul
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(psi : ContinuousMonoidHom G H) (n : ProfiniteKernelSubgroup psi)
(x : ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :
letI : Module (ZCCompletedGroupAlgebra C G)
(ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom)Kernel group-like source scalars act trivially on the separated module after scalar restriction along \(\mathbb{Z}_C\llbracket G\rrbracket \to \mathbb{Z}_C\llbracket H\rrbracket\).
Show proof
Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
zcGroupLike C G n.1 • x = x := by
letI : Module (ZCCompletedGroupAlgebra C G)
(ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
change zcCompletedGroupAlgebraMap C hC psi (zcGroupLike C G n.1) • x = x
rw [zcCompletedGroupAlgebraMap_groupLike]
have hn : psi n.1 = 1 := by
exact MonoidHom.mem_ker.mp
(show n.1 ∈ psi.toMonoidHom.ker from n.2)
rw [hn]
simp only [ContinuousMonoidHom.coe_toMonoidHom, map_one, one_smul]Proof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem zcSeparatedCompletedDifferentialModule_source_map_smul
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(psi : ContinuousMonoidHom G H)
(a : ZCCompletedGroupAlgebra C G)
(x : ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :
letI : Module (ZCCompletedGroupAlgebra C G)
(ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom)Source scalar restriction on the separated module is exactly target scalar multiplication after applying the completed group-algebra map.
Show proof
Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
zcCompletedGroupAlgebraMap C hC psi a • x = a • x := by
letI : Module (ZCCompletedGroupAlgebra C G)
(ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
rflProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem zcSeparatedCompletedDifferentialModule_source_kernel_sub_one_smul
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(psi : ContinuousMonoidHom G H) (n : ProfiniteKernelSubgroup psi)
(x : ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :
letI : Module (ZCCompletedGroupAlgebra C G)
(ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom)Elements of the source kernel augmentation ideal act trivially on the separated module after restricting scalars along the source map.
Show proof
Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
(zcGroupLike C G n.1 - 1) • x = 0 := by
letI : Module (ZCCompletedGroupAlgebra C G)
(ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
calc
(zcGroupLike C G n.1 - 1) • x =
zcGroupLike C G n.1 • x - x := by
rw [sub_smul, one_smul]
_ = 0 := by
rw [zcSeparatedCompletedDifferentialModule_source_kernel_groupLike_smul C hC psi n x]
simp only [ContinuousMonoidHom.coe_toMonoidHom, sub_self]Proof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem zcDiffModuleIdToZCSepDiffModule_kernel_sub_one_smul
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(psi : ContinuousMonoidHom G H) (n : ProfiniteKernelSubgroup psi)
(x : ZCCompletedDifferentialModule C (MonoidHom.id G)) :
zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule C hC psi
((zcGroupLike C G n.1 - 1) • x) = 0The identity-source lift to the separated module kills source kernel augmentation generators after scalar multiplication.
Show proof
by
letI : Module (ZCCompletedGroupAlgebra C G)
(ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
rw [map_smul]
exact
zcSeparatedCompletedDifferentialModule_source_kernel_sub_one_smul
C hC psi n
(zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule
C hC psi x)Proof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□noncomputable def
stdAugIdealToZCSepDiffOfBoundaryKernel
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(psi : ContinuousMonoidHom G H)
(hker :
∀ x : ZCCompletedDifferentialModule C (MonoidHom.id G),
zcToStdAugIdeal C G (MonoidHom.id G) x = 0 →
zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule
C hC psi x = 0) :
letI : Module (ZCCompletedGroupAlgebra C G)
(ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
zcCompletedGroupAlgebraStandardAugmentationIdeal C G →ₗ[ZCCompletedGroupAlgebra C G]
ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom := by
letI : Module (ZCCompletedGroupAlgebra C G)
(ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
let f :=
zcToStdAugIdeal C G (MonoidHom.id G)
let L :=
zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule C hC psi
have hf : Function.Surjective f := by
exact
zcToStdAugIdeal_surjective_of_surjective
C G (MonoidHom.id G) (fun g => ⟨g, rfl⟩)
have hker_le : LinearMap.ker f ≤ LinearMap.ker L := by
intro x hx
rw [LinearMap.mem_ker] at hx ⊢
exact hker x hx
exact
((LinearMap.ker f).liftQ L hker_le).comp
(f.quotKerEquivOfSurjective hf).symm.toLinearMapThe auxiliary Fox-coordinate identity is verified after projection to the corresponding finite quotient and coefficient stage.
noncomputable def
stdAugIdealToZCSepDiff
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(psi : ContinuousMonoidHom G H) :
letI : Module (ZCCompletedGroupAlgebra C G)
(ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
zcCompletedGroupAlgebraStandardAugmentationIdeal C G →ₗ[ZCCompletedGroupAlgebra C G]
ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom :=
stdAugIdealToZCSepDiffOfBoundaryKernel
C hC psi
(zcDiffModuleIdToZCSepDiffModule_eq_zero_of_zcToStandard_eq_zero
C hC psi)
@[simp 900]The auxiliary Fox-coordinate identity is verified after projection to the corresponding finite quotient and coefficient stage.
theorem
stdAugIdealToZCSepDiffOfBoundaryKernel_comp_zcToStdAugIdeal
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(psi : ContinuousMonoidHom G H)
(hker :
∀ x : ZCCompletedDifferentialModule C (MonoidHom.id G),
zcToStdAugIdeal C G (MonoidHom.id G) x = 0 →
zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule
C hC psi x = 0) :
letI : Module (ZCCompletedGroupAlgebra C G)
(ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom)The auxiliary Fox-coordinate identity is verified after projection to the corresponding finite quotient and coefficient stage.
Show proof
Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
(stdAugIdealToZCSepDiffOfBoundaryKernel
C hC psi hker).comp
(zcToStdAugIdeal C G (MonoidHom.id G)) =
zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule C hC psi := by
letI : Module (ZCCompletedGroupAlgebra C G)
(ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
let f :=
zcToStdAugIdeal C G (MonoidHom.id G)
let L :=
zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule C hC psi
have hf : Function.Surjective f := by
exact
zcToStdAugIdeal_surjective_of_surjective
C G (MonoidHom.id G) (fun g => ⟨g, rfl⟩)
have hker_le : LinearMap.ker f ≤ LinearMap.ker L := by
intro x hx
rw [LinearMap.mem_ker] at hx ⊢
exact hker x hx
apply LinearMap.ext
intro x
change
(((LinearMap.ker f).liftQ L hker_le).comp
(f.quotKerEquivOfSurjective hf).symm.toLinearMap).comp f x = L x
rw [LinearMap.comp_apply, LinearMap.comp_apply]
have hsymm :
(f.quotKerEquivOfSurjective hf).symm.toLinearMap (f x) =
Submodule.Quotient.mk x := by
exact LinearMap.quotKerEquivOfSurjective_symm_apply (f := f) hf x
rw [hsymm, Submodule.liftQ_apply]
@[simp]Proof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem
stdAugIdealToZCSepDiff_comp_zcToStdAugIdeal
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(psi : ContinuousMonoidHom G H) :
letI : Module (ZCCompletedGroupAlgebra C G)
(ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom)The auxiliary Fox-coordinate identity is verified after projection to the corresponding finite quotient and coefficient stage.
Show proof
Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
(stdAugIdealToZCSepDiff
C hC psi).comp
(zcToStdAugIdeal C G (MonoidHom.id G)) =
zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule C hC psi := by
exact
stdAugIdealToZCSepDiffOfBoundaryKernel_comp_zcToStdAugIdeal
C hC psi
(zcDiffModuleIdToZCSepDiffModule_eq_zero_of_zcToStandard_eq_zero
C hC psi)Proof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem
stdAugIdealToZCSepDiff_stageBoundary_stageProj
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(psi : ContinuousMonoidHom G H)
(i : ZCCompletedDifferentialModuleIndex C psi.toMonoidHom)
(s : zcCompletedGroupAlgebraStandardAugmentationIdeal C G) :
zcCompletedDifferentialModuleStageBoundary C psi.toMonoidHom i
(zcSeparatedCompletedDifferentialModuleStageProjectionAdd C psi.toMonoidHom i
(stdAugIdealToZCSepDiff
C hC psi s)) =
zcCompletedGroupAlgebraProjection C H i.target
(zcCompletedGroupAlgebraMap C hC psi
(s : ZCCompletedGroupAlgebra C G))The auxiliary Fox-coordinate identity is verified after projection to the corresponding finite quotient and coefficient stage.
Show proof
by
letI : Module (ZCCompletedGroupAlgebra C G)
(ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
let f :=
zcToStdAugIdeal C G (MonoidHom.id G)
let M :=
stdAugIdealToZCSepDiff
C hC psi
let L :=
zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule C hC psi
have hf : Function.Surjective f := by
exact
zcToStdAugIdeal_surjective_of_surjective
C G (MonoidHom.id G) (fun g => ⟨g, rfl⟩)
rcases hf s with ⟨x, hx⟩
rw [← hx]
have hcomp :=
congrArg (fun F => F x)
(stdAugIdealToZCSepDiff_comp_zcToStdAugIdeal
C hC psi)
change M (f x) = L x at hcomp
rw [hcomp]
rw [zcDiffModuleIdToZCSepDiffModule_stageProj]
rw [zcCompletedDifferentialModuleStageBoundary_identitySourceStageToStage]
rw [zcCompletedDifferentialModuleIdentitySourceStageBoundary_stageProj]
let idIndex := zcCompletedDifferentialModuleIdentitySourceIndex C psi.toMonoidHom i
have hmap :=
zcCompletedDifferentialModuleIdentitySourceStageRingHom_projection_map
C hC psi i (f x : ZCCompletedGroupAlgebra C G)
simpa [f, idIndex] using hmapProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem
stdAugIdealToZCSepDiff_stageProj_eq_of_standardProj_eq
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(psi : ContinuousMonoidHom G H)
(i : ZCCompletedDifferentialModuleIndex C psi.toMonoidHom)
{s t : zcCompletedGroupAlgebraStandardAugmentationIdeal C G}
(hst :
zcCompletedGroupAlgebraStandardAugmentationIdealProjection C
(zcCompletedDifferentialModuleIdentitySourceIndex C psi.toMonoidHom i).target s =
zcCompletedGroupAlgebraStandardAugmentationIdealProjection C
(zcCompletedDifferentialModuleIdentitySourceIndex C psi.toMonoidHom i).target t) :
zcSeparatedCompletedDifferentialModuleStageProjectionAdd C psi.toMonoidHom i
(stdAugIdealToZCSepDiff
C hC psi s) =
zcSeparatedCompletedDifferentialModuleStageProjectionAdd C psi.toMonoidHom i
(stdAugIdealToZCSepDiff
C hC psi t)The auxiliary Fox-coordinate identity is verified after projection to the corresponding finite quotient and coefficient stage.
Show proof
by
letI : Module (ZCCompletedGroupAlgebra C G)
(ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
let f :=
zcToStdAugIdeal C G (MonoidHom.id G)
let M :=
stdAugIdealToZCSepDiff
C hC psi
let L :=
zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule C hC psi
let idIndex := zcCompletedDifferentialModuleIdentitySourceIndex C psi.toMonoidHom i
have hf : Function.Surjective f := by
exact
zcToStdAugIdeal_surjective_of_surjective
C G (MonoidHom.id G) (fun g => ⟨g, rfl⟩)
rcases hf s with ⟨x, hx⟩
rcases hf t with ⟨y, hy⟩
have hxyProjection :
zcCompletedGroupAlgebraStandardAugmentationIdealProjection C idIndex.target (f x) =
zcCompletedGroupAlgebraStandardAugmentationIdealProjection C idIndex.target (f y) := by
simpa [idIndex, hx, hy] using hst
have hcomp_x :=
congrArg (fun F => F x)
(stdAugIdealToZCSepDiff_comp_zcToStdAugIdeal
C hC psi)
have hcomp_y :=
congrArg (fun F => F y)
(stdAugIdealToZCSepDiff_comp_zcToStdAugIdeal
C hC psi)
change M (f x) = L x at hcomp_x
change M (f y) = L y at hcomp_y
rw [← hx, ← hy, hcomp_x, hcomp_y]
rw [zcDiffModuleIdToZCSepDiffModule_stageProj,
zcDiffModuleIdToZCSepDiffModule_stageProj]
have hsource :
zcCompletedDifferentialModuleStageProjection C (MonoidHom.id G) idIndex x =
zcCompletedDifferentialModuleStageProjection C (MonoidHom.id G) idIndex y := by
apply sub_eq_zero.mp
have hzero :
zcCompletedDifferentialModuleStageProjection C (MonoidHom.id G) idIndex (x - y) = 0 :=
zcDiffModuleIdentitySourceStageProj_eq_zero_of_boundary_eq_zero
C psi.toMonoidHom i (x - y) (by
rw [map_sub]
rw [map_sub]
rw [zcCompletedDifferentialModuleIdentitySourceStageBoundary_stageProj,
zcCompletedDifferentialModuleIdentitySourceStageBoundary_stageProj]
rw [hxyProjection, sub_self])
simpa [map_sub] using hzero
rw [hsource]Proof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem
continuous_stdAugIdealToZCSepDiff_stageProj
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(psi : ContinuousMonoidHom G H)
(i : ZCCompletedDifferentialModuleIndex C psi.toMonoidHom) :
Continuous
(fun s : zcCompletedGroupAlgebraStandardAugmentationIdeal C G =>
zcSeparatedCompletedDifferentialModuleStageProjectionAdd C psi.toMonoidHom i
(stdAugIdealToZCSepDiff
C hC psi s))The auxiliary Fox-coordinate identity is verified after projection to the corresponding finite quotient and coefficient stage.
Show proof
by
let idIndex := zcCompletedDifferentialModuleIdentitySourceIndex C psi.toMonoidHom i
let p :=
zcCompletedGroupAlgebraStandardAugmentationIdealProjection C idIndex.target
have hsurj : Function.Surjective p :=
zcCompletedGroupAlgebraStandardAugmentationIdealProjection_surjective C idIndex.target
let F : zcCompletedGroupAlgebraStageAugmentationIdeal C G idIndex.target →
ZCCompletedDifferentialModuleStage C psi.toMonoidHom i := fun y =>
zcSeparatedCompletedDifferentialModuleStageProjectionAdd C psi.toMonoidHom i
(stdAugIdealToZCSepDiff
C hC psi (Classical.choose (hsurj y)))
have hfactor :
(fun s : zcCompletedGroupAlgebraStandardAugmentationIdeal C G =>
zcSeparatedCompletedDifferentialModuleStageProjectionAdd C psi.toMonoidHom i
(stdAugIdealToZCSepDiff
C hC psi s)) =
fun s => F (p s) := by
funext s
exact
stdAugIdealToZCSepDiff_stageProj_eq_of_standardProj_eq
C hC psi i
(by
dsimp [p, F]
exact (Classical.choose_spec (hsurj (p s))).symm)
rw [hfactor]
haveI : DiscreteTopology (zcCompletedGroupAlgebraStageAugmentationIdeal C G idIndex.target) := by
infer_instance
have hF : Continuous F :=
continuous_of_discreteTopology
exact hF.comp (continuous_zcCompletedGroupAlgebraStandardAugmentationIdealProjection C idIndex.target)Proof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem
continuous_stdAugIdealToZCSepDiff_stageProjProduct
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(psi : ContinuousMonoidHom G H) :
Continuous
(fun s : zcCompletedGroupAlgebraStandardAugmentationIdeal C G =>
zcSeparatedCompletedDifferentialModuleStageProjectionProduct C psi.toMonoidHom
(stdAugIdealToZCSepDiff
C hC psi s))The auxiliary Fox-coordinate identity is verified after projection to the corresponding finite quotient and coefficient stage.
Show proof
by
exact continuous_pi fun i =>
continuous_stdAugIdealToZCSepDiff_stageProj
C hC psi iProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem
continuous_stdAugIdealToZCSepDiff
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) :
letI : TopologicalSpace (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom)The auxiliary Fox-coordinate identity is verified after projection to the corresponding finite quotient and coefficient stage.
Show proof
zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom
@Continuous
(zcCompletedGroupAlgebraStandardAugmentationIdeal C G)
(ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom)
inferInstance
(zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom)
(stdAugIdealToZCSepDiff
C hC psi) := by
letI : Nonempty (ZCCompletedDifferentialModuleIndex C psi.toMonoidHom) :=
nonempty_zcCompletedDifferentialModuleIndex C hC psi
rw [zcSepDiffModuleNaturalTopology_eq_induced_stageProjProduct
C psi.toMonoidHom (directed_zcCompletedDifferentialModuleIndex C hForm hC psi)]
rw [continuous_induced_rng]
exact
continuous_stdAugIdealToZCSepDiff_stageProjProduct
C hC psi
@[simp 900]Proof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem
stdAugIdealToZCSepDiffOfBoundaryKernel_boundary
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(psi : ContinuousMonoidHom G H)
(hker :
∀ x : ZCCompletedDifferentialModule C (MonoidHom.id G),
zcToStdAugIdeal C G (MonoidHom.id G) x = 0 →
zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule
C hC psi x = 0)
(g : G) :
stdAugIdealToZCSepDiffOfBoundaryKernel
C hC psi hker
⟨zcCompletedGroupAlgebraBoundary C (MonoidHom.id G) g,
zcCompletedGroupAlgebraBoundary_mem_standardAugmentationIdeal
C G (MonoidHom.id G) g⟩ =
zcSeparatedUniversalDifferential C psi.toMonoidHom gThe auxiliary Fox-coordinate identity is verified after projection to the corresponding finite quotient and coefficient stage.
Show proof
by
letI : Module (ZCCompletedGroupAlgebra C G)
(ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
have hcomp :=
congrArg
(fun f =>
f (zcUniversalDifferential C (MonoidHom.id G) g))
(stdAugIdealToZCSepDiffOfBoundaryKernel_comp_zcToStdAugIdeal
C hC psi hker)
calc
stdAugIdealToZCSepDiffOfBoundaryKernel
C hC psi hker
⟨zcCompletedGroupAlgebraBoundary C (MonoidHom.id G) g,
zcCompletedGroupAlgebraBoundary_mem_standardAugmentationIdeal
C G (MonoidHom.id G) g⟩ =
zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule
C hC psi (zcUniversalDifferential C (MonoidHom.id G) g) := by
simpa [zcToStdAugIdeal,
zcCompletedGroupAlgebraBoundaryToStandardAugmentationIdeal] using hcomp
_ = zcSeparatedUniversalDifferential C psi.toMonoidHom g := by
exact
zcDiffModuleIdToZCSepDiffModule_universal
C hC psi g
@[simp]Proof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem
stdAugIdealToZCSepDiff_boundary
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(psi : ContinuousMonoidHom G H)
(g : G) :
stdAugIdealToZCSepDiff
C hC psi
⟨zcCompletedGroupAlgebraBoundary C (MonoidHom.id G) g,
zcCompletedGroupAlgebraBoundary_mem_standardAugmentationIdeal
C G (MonoidHom.id G) g⟩ =
zcSeparatedUniversalDifferential C psi.toMonoidHom gThe auxiliary Fox-coordinate identity is verified after projection to the corresponding finite quotient and coefficient stage.
Show proof
by
exact
stdAugIdealToZCSepDiffOfBoundaryKernel_boundary
C hC psi
(zcDiffModuleIdToZCSepDiffModule_eq_zero_of_zcToStandard_eq_zero
C hC psi) gProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem
stdAugIdealToZCSepDiffOfBoundaryKernel_kernel_generator_smul
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(psi : ContinuousMonoidHom G H)
(hker :
∀ x : ZCCompletedDifferentialModule C (MonoidHom.id G),
zcToStdAugIdeal C G (MonoidHom.id G) x = 0 →
zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule
C hC psi x = 0)
(n : ProfiniteKernelSubgroup psi)
(s : zcCompletedGroupAlgebraStandardAugmentationIdeal C G) :
stdAugIdealToZCSepDiffOfBoundaryKernel
C hC psi hker ((zcGroupLike C G n.1 - 1) • s) = 0The auxiliary Fox-coordinate identity is verified after projection to the corresponding finite quotient and coefficient stage.
Show proof
by
letI : Module (ZCCompletedGroupAlgebra C G)
(ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
let f :=
zcToStdAugIdeal C G (MonoidHom.id G)
let M :=
stdAugIdealToZCSepDiffOfBoundaryKernel
C hC psi hker
let L :=
zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule C hC psi
have hf : Function.Surjective f := by
exact
zcToStdAugIdeal_surjective_of_surjective
C G (MonoidHom.id G) (fun g => ⟨g, rfl⟩)
rcases hf s with ⟨y, hy⟩
have hcomp :=
congrArg
(fun F =>
F ((zcGroupLike C G n.1 - 1) • y))
(stdAugIdealToZCSepDiffOfBoundaryKernel_comp_zcToStdAugIdeal
C hC psi hker)
have harg :
(zcGroupLike C G n.1 - 1) • s =
f ((zcGroupLike C G n.1 - 1) • y) := by
rw [map_smul, hy]
calc
M ((zcGroupLike C G n.1 - 1) • s) =
M (f ((zcGroupLike C G n.1 - 1) • y)) := by
rw [harg]
_ = L ((zcGroupLike C G n.1 - 1) • y) := by
simpa [M, L, f] using hcomp
_ = 0 := by
exact
zcDiffModuleIdToZCSepDiffModule_kernel_sub_one_smul
C hC psi n yProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem
stdAugIdealToZCSepDiffOfBoundaryKernel_kills_kernelMulStandard
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(psi : ContinuousMonoidHom G H)
(hker :
∀ x : ZCCompletedDifferentialModule C (MonoidHom.id G),
zcToStdAugIdeal C G (MonoidHom.id G) x = 0 →
zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule
C hC psi x = 0)
{x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G}
(hx : x ∈ zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi) :
stdAugIdealToZCSepDiffOfBoundaryKernel
C hC psi hker x = 0The auxiliary Fox-coordinate identity is verified after projection to the corresponding finite quotient and coefficient stage.
Show proof
by
letI : Module (ZCCompletedGroupAlgebra C G)
(ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
let M :=
stdAugIdealToZCSepDiffOfBoundaryKernel
C hC psi hker
rw [zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard] at hx
refine Submodule.span_induction
(p := fun y _ => M y = 0) ?_ ?_ ?_ ?_ hx
· rintro _ ⟨p, rfl⟩
exact
stdAugIdealToZCSepDiffOfBoundaryKernel_kernel_generator_smul
C hC psi hker p.1 p.2
· simp only [ContinuousMonoidHom.coe_toMonoidHom, map_zero, M]
· intro y z _ _ hy hz
rw [map_add, hy, hz, add_zero]
· intro a y _ hy
rw [map_smul, hy, smul_zero]Proof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem
stdAugIdealToZCSepDiff_kills_kernelMulStandard
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(psi : ContinuousMonoidHom G H)
{x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G}
(hx : x ∈ zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi) :
stdAugIdealToZCSepDiff
C hC psi x = 0The auxiliary Fox-coordinate identity is verified after projection to the corresponding finite quotient and coefficient stage.
Show proof
stdAugIdealToZCSepDiffOfBoundaryKernel_kills_kernelMulStandard
C hC psi
(zcDiffModuleIdToZCSepDiffModule_eq_zero_of_zcToStandard_eq_zero
C hC psi) hxProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem
stdAugIdealToZCSepDiffOfBoundaryKernel_kills_kernelMulStandardClosed_of_continuous
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(hker :
∀ x : ZCCompletedDifferentialModule C (MonoidHom.id G),
zcToStdAugIdeal C G (MonoidHom.id G) x = 0 →
zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule
C hC psi x = 0)
(hcont :
letI : TopologicalSpace (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom
@Continuous
(zcCompletedGroupAlgebraStandardAugmentationIdeal C G)
(ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom)
inferInstance
(zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom)
(stdAugIdealToZCSepDiffOfBoundaryKernel
C hC psi hker))
{x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G}
(hx : x ∈ zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen) :
stdAugIdealToZCSepDiffOfBoundaryKernel
C hC psi hker x = 0The auxiliary Fox-coordinate identity is verified after projection to the corresponding finite quotient and coefficient stage.
Show proof
by
letI : TopologicalSpace (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom
let M :=
stdAugIdealToZCSepDiffOfBoundaryKernel
C hC psi hker
have hxcl :
x ∈ closure
((zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi :
Set (zcCompletedGroupAlgebraStandardAugmentationIdeal C G))) := by
rw [closure_zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard_eq_closed
C hC hForm psi hpsi hfopen]
exact hx
have hclosed_preimage :
IsClosed
(M ⁻¹'
({0} : Set (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom))) := by
exact
(isClosed_zero_zcSeparatedCompletedDifferentialModuleNaturalTopology
C psi.toMonoidHom).preimage hcont
have hsubset :
((zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi :
Set (zcCompletedGroupAlgebraStandardAugmentationIdeal C G))) ⊆
M ⁻¹' ({0} : Set (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom)) := by
intro y hy
exact
stdAugIdealToZCSepDiffOfBoundaryKernel_kills_kernelMulStandard
C hC psi hker hy
exact closure_minimal hsubset hclosed_preimage hxclProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem
stdAugIdealToZCSepDiff_kills_kernelMulStandardClosed_of_continuous
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(hcont :
letI : TopologicalSpace (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom
@Continuous
(zcCompletedGroupAlgebraStandardAugmentationIdeal C G)
(ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom)
inferInstance
(zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom)
(stdAugIdealToZCSepDiff
C hC psi))
{x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G}
(hx : x ∈ zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen) :
stdAugIdealToZCSepDiff
C hC psi x = 0The auxiliary Fox-coordinate identity is verified after projection to the corresponding finite quotient and coefficient stage.
Show proof
by
have hcont' :
letI : TopologicalSpace (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom
@Continuous
(zcCompletedGroupAlgebraStandardAugmentationIdeal C G)
(ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom)
inferInstance
(zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom)
(stdAugIdealToZCSepDiffOfBoundaryKernel
C hC psi
(zcDiffModuleIdToZCSepDiffModule_eq_zero_of_zcToStandard_eq_zero
C hC psi)) := by
simpa [stdAugIdealToZCSepDiff] using hcont
exact
stdAugIdealToZCSepDiffOfBoundaryKernel_kills_kernelMulStandardClosed_of_continuous
C hC hForm psi hpsi hfopen
(zcDiffModuleIdToZCSepDiffModule_eq_zero_of_zcToStandard_eq_zero
C hC psi) hcont' hxProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem
stdAugIdealToZCSepDiff_kills_kernelMulStandardClosed
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
{x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G}
(hx : x ∈ zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen) :
stdAugIdealToZCSepDiff
C hC psi x = 0The auxiliary Fox-coordinate identity is verified after projection to the corresponding finite quotient and coefficient stage.
Show proof
stdAugIdealToZCSepDiff_kills_kernelMulStandardClosed_of_continuous
C hC hForm psi hpsi hfopen
(continuous_stdAugIdealToZCSepDiff
C hC hForm psi) hxProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□noncomputable def
kerAugIdealQuotToZCSepDiffOfBoundaryKernelOfClosedKill
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(hker :
∀ x : ZCCompletedDifferentialModule C (MonoidHom.id G),
zcToStdAugIdeal C G (MonoidHom.id G) x = 0 →
zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule
C hC psi x = 0)
(hclosed_kill :
∀ x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G,
x ∈ zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen →
stdAugIdealToZCSepDiffOfBoundaryKernel
C hC psi hker x = 0) :
letI : Module (ZCCompletedGroupAlgebra C G)
(ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen
→ₗ[ZCCompletedGroupAlgebra C G]
ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom := by
letI : Module (ZCCompletedGroupAlgebra C G)
(ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
let M :=
stdAugIdealToZCSepDiffOfBoundaryKernel
C hC psi hker
exact
(zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen).liftQ M
(by
intro x hx
rw [LinearMap.mem_ker]
exact hclosed_kill x hx)
@[simp 900]The auxiliary Fox-coordinate identity is verified after projection to the corresponding finite quotient and coefficient stage.
theorem
kerAugIdealQuotToZCSepDiffOfBoundaryKernelOfClosedKill_boundary
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(hker :
∀ x : ZCCompletedDifferentialModule C (MonoidHom.id G),
zcToStdAugIdeal C G (MonoidHom.id G) x = 0 →
zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule
C hC psi x = 0)
(hclosed_kill :
∀ x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G,
x ∈ zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen →
stdAugIdealToZCSepDiffOfBoundaryKernel
C hC psi hker x = 0)
(g : G) :
kerAugIdealQuotToZCSepDiffOfBoundaryKernelOfClosedKill
C hC hForm psi hpsi hfopen hker hclosed_kill
(Submodule.Quotient.mk
(p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen)
⟨zcCompletedGroupAlgebraBoundary C (MonoidHom.id G) g,
zcCompletedGroupAlgebraBoundary_mem_standardAugmentationIdeal
C G (MonoidHom.id G) g⟩) =
zcSeparatedUniversalDifferential C psi.toMonoidHom gThe auxiliary Fox-coordinate identity is verified after projection to the corresponding finite quotient and coefficient stage.
Show proof
by
letI : Module (ZCCompletedGroupAlgebra C G)
(ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
rw [kerAugIdealQuotToZCSepDiffOfBoundaryKernelOfClosedKill,
Submodule.liftQ_apply]
exact
stdAugIdealToZCSepDiffOfBoundaryKernel_boundary
C hC psi hker gProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□noncomputable def
kerAugIdealQuotToZCSepDiffLinearOfBoundaryKernelOfClosedKill
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(hker :
∀ x : ZCCompletedDifferentialModule C (MonoidHom.id G),
zcToStdAugIdeal C G (MonoidHom.id G) x = 0 →
zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule
C hC psi x = 0)
(hclosed_kill :
∀ x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G,
x ∈ zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen →
stdAugIdealToZCSepDiffOfBoundaryKernel
C hC psi hker x = 0) :
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen
→ₗ[ZCCompletedGroupAlgebra C H]
ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom := by
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
letI : Module (ZCCompletedGroupAlgebra C G)
(ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
let Q :=
kerAugIdealQuotToZCSepDiffOfBoundaryKernelOfClosedKill
C hC hForm psi hpsi hfopen hker hclosed_kill
refine
{ toFun := Q
map_add' := by
intro x y
exact map_add Q x y
map_smul' := by
intro a x
change Q
(zcCompletedGroupAlgebraTargetLiftOfSurjective C hC hForm psi hpsi a • x) =
a • Q x
rw [map_smul]
symm
calc
a • Q x =
zcCompletedGroupAlgebraMap C hC psi
(zcCompletedGroupAlgebraTargetLiftOfSurjective C hC hForm psi hpsi a) •
Q x := by
rw [zcCompletedGroupAlgebraMap_targetLiftOfSurjective]
_ =
zcCompletedGroupAlgebraTargetLiftOfSurjective C hC hForm psi hpsi a •
Q x := by
exact
zcSeparatedCompletedDifferentialModule_source_map_smul
C hC psi
(zcCompletedGroupAlgebraTargetLiftOfSurjective C hC hForm psi hpsi a)
(Q x) }
@[simp 900]The auxiliary Fox-coordinate identity is verified after projection to the corresponding finite quotient and coefficient stage.
theorem
kerAugIdealQuotToZCSepDiffLinearOfBoundaryKernelOfClosedKill_boundary
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(hker :
∀ x : ZCCompletedDifferentialModule C (MonoidHom.id G),
zcToStdAugIdeal C G (MonoidHom.id G) x = 0 →
zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule
C hC psi x = 0)
(hclosed_kill :
∀ x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G,
x ∈ zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen →
stdAugIdealToZCSepDiffOfBoundaryKernel
C hC psi hker x = 0)
(g : G) :
kerAugIdealQuotToZCSepDiffLinearOfBoundaryKernelOfClosedKill
C hC hForm psi hpsi hfopen hker hclosed_kill
(Submodule.Quotient.mk
(p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen)
⟨zcCompletedGroupAlgebraBoundary C (MonoidHom.id G) g,
zcCompletedGroupAlgebraBoundary_mem_standardAugmentationIdeal
C G (MonoidHom.id G) g⟩) =
zcSeparatedUniversalDifferential C psi.toMonoidHom gThe auxiliary Fox-coordinate identity is verified after projection to the corresponding finite quotient and coefficient stage.
Show proof
by
exact
kerAugIdealQuotToZCSepDiffOfBoundaryKernelOfClosedKill_boundary
C hC hForm psi hpsi hfopen hker hclosed_kill gProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□noncomputable def
kerAugIdealQuotToZCSepDiffLinearOfBoundaryKernelOfContStdMap
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(hker :
∀ x : ZCCompletedDifferentialModule C (MonoidHom.id G),
zcToStdAugIdeal C G (MonoidHom.id G) x = 0 →
zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule
C hC psi x = 0)
(hcont :
letI : TopologicalSpace (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom
@Continuous
(zcCompletedGroupAlgebraStandardAugmentationIdeal C G)
(ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom)
inferInstance
(zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom)
(stdAugIdealToZCSepDiffOfBoundaryKernel
C hC psi hker)) :
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen
→ₗ[ZCCompletedGroupAlgebra C H]
ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom :=
kerAugIdealQuotToZCSepDiffLinearOfBoundaryKernelOfClosedKill
C hC hForm psi hpsi hfopen hker
(fun _ hx =>
stdAugIdealToZCSepDiffOfBoundaryKernel_kills_kernelMulStandardClosed_of_continuous
C hC hForm psi hpsi hfopen hker hcont hx)
@[simp 900]The auxiliary Fox-coordinate identity is verified after projection to the corresponding finite quotient and coefficient stage.
theorem
kerAugIdealQuotToZCSepDiffLinearOfBoundaryKernelOfContStdMap_boundary
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(hker :
∀ x : ZCCompletedDifferentialModule C (MonoidHom.id G),
zcToStdAugIdeal C G (MonoidHom.id G) x = 0 →
zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule
C hC psi x = 0)
(hcont :
letI : TopologicalSpace (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom
@Continuous
(zcCompletedGroupAlgebraStandardAugmentationIdeal C G)
(ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom)
inferInstance
(zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom)
(stdAugIdealToZCSepDiffOfBoundaryKernel
C hC psi hker))
(g : G) :
kerAugIdealQuotToZCSepDiffLinearOfBoundaryKernelOfContStdMap
C hC hForm psi hpsi hfopen hker hcont
(Submodule.Quotient.mk
(p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen)
⟨zcCompletedGroupAlgebraBoundary C (MonoidHom.id G) g,
zcCompletedGroupAlgebraBoundary_mem_standardAugmentationIdeal
C G (MonoidHom.id G) g⟩) =
zcSeparatedUniversalDifferential C psi.toMonoidHom gThe auxiliary Fox-coordinate identity is verified after projection to the corresponding finite quotient and coefficient stage.
Show proof
by
exact
kerAugIdealQuotToZCSepDiffLinearOfBoundaryKernelOfClosedKill_boundary
C hC hForm psi hpsi hfopen hker
(fun x hx =>
stdAugIdealToZCSepDiffOfBoundaryKernel_kills_kernelMulStandardClosed_of_continuous
C hC hForm psi hpsi hfopen hker hcont hx)
gProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□noncomputable def
kerAugIdealQuotToZCSepDiffOfClosedKill
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(hclosed_kill :
∀ x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G,
x ∈ zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen →
stdAugIdealToZCSepDiff
C hC psi x = 0) :
letI : Module (ZCCompletedGroupAlgebra C G)
(ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen
→ₗ[ZCCompletedGroupAlgebra C G]
ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom :=
kerAugIdealQuotToZCSepDiffOfBoundaryKernelOfClosedKill
C hC hForm psi hpsi hfopen
(zcDiffModuleIdToZCSepDiffModule_eq_zero_of_zcToStandard_eq_zero
C hC psi)
(fun x hx => by
simpa [stdAugIdealToZCSepDiff]
using hclosed_kill x hx)
@[simp 900]The auxiliary Fox-coordinate identity is verified after projection to the corresponding finite quotient and coefficient stage.
theorem
kerAugIdealQuotToZCSepDiffOfClosedKill_boundary
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(hclosed_kill :
∀ x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G,
x ∈ zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen →
stdAugIdealToZCSepDiff
C hC psi x = 0)
(g : G) :
kerAugIdealQuotToZCSepDiffOfClosedKill
C hC hForm psi hpsi hfopen hclosed_kill
(Submodule.Quotient.mk
(p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen)
⟨zcCompletedGroupAlgebraBoundary C (MonoidHom.id G) g,
zcCompletedGroupAlgebraBoundary_mem_standardAugmentationIdeal
C G (MonoidHom.id G) g⟩) =
zcSeparatedUniversalDifferential C psi.toMonoidHom gThe auxiliary Fox-coordinate identity is verified after projection to the corresponding finite quotient and coefficient stage.
Show proof
by
exact
kerAugIdealQuotToZCSepDiffOfBoundaryKernelOfClosedKill_boundary
C hC hForm psi hpsi hfopen
(zcDiffModuleIdToZCSepDiffModule_eq_zero_of_zcToStandard_eq_zero
C hC psi)
(fun x hx => by
simpa [stdAugIdealToZCSepDiff]
using hclosed_kill x hx) gProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem
continuous_kerAugIdealQuotToZCSepDiffOfClosedKill
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(hclosed_kill :
∀ x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G,
x ∈ zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen →
stdAugIdealToZCSepDiff
C hC psi x = 0)
(hcont :
letI : TopologicalSpace (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom
@Continuous
(zcCompletedGroupAlgebraStandardAugmentationIdeal C G)
(ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom)
inferInstance
(zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom)
(stdAugIdealToZCSepDiff
C hC psi)) :
letI : TopologicalSpace (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom)The auxiliary Fox-coordinate identity is verified after projection to the corresponding finite quotient and coefficient stage.
Show proof
zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom
Continuous
(kerAugIdealQuotToZCSepDiffOfClosedKill
C hC hForm psi hpsi hfopen hclosed_kill) := by
letI : TopologicalSpace (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom
rw [continuous_kernelAugmentationIdealClosedQuotient_iff_comp_mkQ
C hC hForm psi hpsi hfopen]
have hcomp :
(fun x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G =>
kerAugIdealQuotToZCSepDiffOfClosedKill
C hC hForm psi hpsi hfopen hclosed_kill
((zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen).mkQ x)) =
stdAugIdealToZCSepDiff
C hC psi := by
funext x
simp only [ContinuousMonoidHom.coe_toMonoidHom, kerAugIdealQuotToZCSepDiffOfClosedKill,
kerAugIdealQuotToZCSepDiffOfBoundaryKernelOfClosedKill, Submodule.mkQ_apply, Submodule.liftQ_apply,
stdAugIdealToZCSepDiff]
rw [hcomp]
exact hcontProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□noncomputable def
kerAugIdealQuotToZCSepDiffLinearOfClosedKill
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(hclosed_kill :
∀ x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G,
x ∈ zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen →
stdAugIdealToZCSepDiff
C hC psi x = 0) :
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen
→ₗ[ZCCompletedGroupAlgebra C H]
ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom :=
kerAugIdealQuotToZCSepDiffLinearOfBoundaryKernelOfClosedKill
C hC hForm psi hpsi hfopen
(zcDiffModuleIdToZCSepDiffModule_eq_zero_of_zcToStandard_eq_zero
C hC psi)
(fun x hx => by
simpa [stdAugIdealToZCSepDiff]
using hclosed_kill x hx)
@[simp 900]The auxiliary Fox-coordinate identity is verified after projection to the corresponding finite quotient and coefficient stage.
theorem
kerAugIdealQuotToZCSepDiffLinearOfClosedKill_boundary
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(hclosed_kill :
∀ x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G,
x ∈ zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen →
stdAugIdealToZCSepDiff
C hC psi x = 0)
(g : G) :
kerAugIdealQuotToZCSepDiffLinearOfClosedKill
C hC hForm psi hpsi hfopen hclosed_kill
(Submodule.Quotient.mk
(p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen)
⟨zcCompletedGroupAlgebraBoundary C (MonoidHom.id G) g,
zcCompletedGroupAlgebraBoundary_mem_standardAugmentationIdeal
C G (MonoidHom.id G) g⟩) =
zcSeparatedUniversalDifferential C psi.toMonoidHom gThe auxiliary Fox-coordinate identity is verified after projection to the corresponding finite quotient and coefficient stage.
Show proof
kerAugIdealQuotToZCSepDiffOfClosedKill_boundary
C hC hForm psi hpsi hfopen hclosed_kill gProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□noncomputable def
kerAugIdealQuotToZCSepDiffLinearOfContStdMap
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(hcont :
letI : TopologicalSpace (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom
@Continuous
(zcCompletedGroupAlgebraStandardAugmentationIdeal C G)
(ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom)
inferInstance
(zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom)
(stdAugIdealToZCSepDiff
C hC psi)) :
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen
→ₗ[ZCCompletedGroupAlgebra C H]
ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom :=
kerAugIdealQuotToZCSepDiffLinearOfClosedKill
C hC hForm psi hpsi hfopen
(fun _ hx =>
stdAugIdealToZCSepDiff_kills_kernelMulStandardClosed_of_continuous
C hC hForm psi hpsi hfopen hcont hx)
@[simp 900]The auxiliary Fox-coordinate identity is verified after projection to the corresponding finite quotient and coefficient stage.
theorem
kerAugIdealQuotToZCSepDiffLinearOfContStdMap_boundary
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(hcont :
letI : TopologicalSpace (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom
@Continuous
(zcCompletedGroupAlgebraStandardAugmentationIdeal C G)
(ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom)
inferInstance
(zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom)
(stdAugIdealToZCSepDiff
C hC psi))
(g : G) :
kerAugIdealQuotToZCSepDiffLinearOfContStdMap
C hC hForm psi hpsi hfopen hcont
(Submodule.Quotient.mk
(p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen)
⟨zcCompletedGroupAlgebraBoundary C (MonoidHom.id G) g,
zcCompletedGroupAlgebraBoundary_mem_standardAugmentationIdeal
C G (MonoidHom.id G) g⟩) =
zcSeparatedUniversalDifferential C psi.toMonoidHom gThe auxiliary Fox-coordinate identity is verified after projection to the corresponding finite quotient and coefficient stage.
Show proof
kerAugIdealQuotToZCSepDiffLinearOfClosedKill_boundary
C hC hForm psi hpsi hfopen
(fun _ hx =>
stdAugIdealToZCSepDiff_kills_kernelMulStandardClosed_of_continuous
C hC hForm psi hpsi hfopen hcont hx) gProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem
continuous_kerAugIdealQuotToZCSepDiffLinearOfContStdMap
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(hcont :
letI : TopologicalSpace (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom
@Continuous
(zcCompletedGroupAlgebraStandardAugmentationIdeal C G)
(ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom)
inferInstance
(zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom)
(stdAugIdealToZCSepDiff
C hC psi)) :
letI : TopologicalSpace (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom)The auxiliary Fox-coordinate identity is verified after projection to the corresponding finite quotient and coefficient stage.
Show proof
zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom
Continuous
(kerAugIdealQuotToZCSepDiffLinearOfContStdMap
C hC hForm psi hpsi hfopen hcont) := by
letI : TopologicalSpace (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom
simpa [kerAugIdealQuotToZCSepDiffLinearOfContStdMap,
kerAugIdealQuotToZCSepDiffLinearOfClosedKill]
using
continuous_kerAugIdealQuotToZCSepDiffOfClosedKill
C hC hForm psi hpsi hfopen
(fun _ hx =>
stdAugIdealToZCSepDiff_kills_kernelMulStandardClosed_of_continuous
C hC hForm psi hpsi hfopen hcont hx)
hcontProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□noncomputable def
kerAugIdealQuotToZCSepDiffLinear
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi) :
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen
→ₗ[ZCCompletedGroupAlgebra C H]
ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom :=
kerAugIdealQuotToZCSepDiffLinearOfContStdMap
C hC hForm psi hpsi hfopen
(continuous_stdAugIdealToZCSepDiff
C hC hForm psi)
@[simp 900]The auxiliary Fox-coordinate identity is verified after projection to the corresponding finite quotient and coefficient stage.
theorem
kerAugIdealQuotToZCSepDiffLinear_boundary
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(g : G) :
kerAugIdealQuotToZCSepDiffLinear
C hC hForm psi hpsi hfopen
(Submodule.Quotient.mk
(p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen)
⟨zcCompletedGroupAlgebraBoundary C (MonoidHom.id G) g,
zcCompletedGroupAlgebraBoundary_mem_standardAugmentationIdeal
C G (MonoidHom.id G) g⟩) =
zcSeparatedUniversalDifferential C psi.toMonoidHom gThe auxiliary Fox-coordinate identity is verified after projection to the corresponding finite quotient and coefficient stage.
Show proof
kerAugIdealQuotToZCSepDiffLinearOfContStdMap_boundary
C hC hForm psi hpsi hfopen
(continuous_stdAugIdealToZCSepDiff
C hC hForm psi) gProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem
continuous_kerAugIdealQuotToZCSepDiffLinear
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi) :
letI : TopologicalSpace (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom)The auxiliary Fox-coordinate identity is verified after projection to the corresponding finite quotient and coefficient stage.
Show proof
zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom
Continuous
(kerAugIdealQuotToZCSepDiffLinear
C hC hForm psi hpsi hfopen) :=
continuous_kerAugIdealQuotToZCSepDiffLinearOfContStdMap
C hC hForm psi hpsi hfopen
(continuous_stdAugIdealToZCSepDiff
C hC hForm psi)Proof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□def zcCompletedDifferentialModuleToKernelAugmentationClosedQuotientOfSurjective
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi) :
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
ZCCompletedDifferentialModule C psi.toMonoidHom →ₗ[ZCCompletedGroupAlgebra C H]
KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen := by
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
exact
zcCompletedDifferentialModuleLift
(A := KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)
C psi.toMonoidHom
(zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
C hC hForm psi hpsi hfopen)
(zcCompletedGASourceBoundaryToKerAugClosedQuot_isTargetCompletedCrossedDiff_of_surj
C hC hForm psi hpsi hfopen)
@[simp 900]The completed universal differential module maps to the closed source augmentation quotient by \(dg \mapsto\) \([g]-1\).
theorem zcDiffToKerAugClosedQuotOfSurj_universal
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi) (g : G) :
zcCompletedDifferentialModuleToKernelAugmentationClosedQuotientOfSurjective
C hC hForm psi hpsi hfopen
(zcUniversalDifferential C psi.toMonoidHom g) =
zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
C hC hForm psi hpsi hfopen gThe closed augmentation-quotient map sends the universal differential to the class of the corresponding augmentation generator.
Show proof
by
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
exact
zcCompletedDifferentialModuleLift_universal
(A := KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)
C psi.toMonoidHom
(zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
C hC hForm psi hpsi hfopen)
(zcCompletedGASourceBoundaryToKerAugClosedQuot_isTargetCompletedCrossedDiff_of_surj
C hC hForm psi hpsi hfopen) gProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem zcDiffToKerAugClosedQuotOfSurj_kills_finiteClosedSubmodule_of_continuous_lift
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
[Nonempty (ZCCompletedDifferentialModuleIndex C psi.toMonoidHom)]
(hdir : Directed (· ≤ ·)
(id : ZCCompletedDifferentialModuleIndex C psi.toMonoidHom →
ZCCompletedDifferentialModuleIndex C psi.toMonoidHom))
(hcont :
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
@Continuous
(CrossedDifferentialPreModule (ZCCompletedGroupAlgebra C H) G)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)
(zcCompletedDifferentialPreModuleNaturalTopology C psi.toMonoidHom)
inferInstance
(crossedDifferentialModuleLiftLinear
(R := ZCCompletedGroupAlgebra C H)
(zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
C hC hForm psi hpsi hfopen)))
{x : CrossedDifferentialPreModule (ZCCompletedGroupAlgebra C H) G}
(hx : x ∈ zcCompletedDifferentialRelationFiniteClosedSubmodule C psi.toMonoidHom) :
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)If the pre-quotient source-boundary lift to the closed augmentation quotient is continuous for the finite-stage pre-module topology, then it kills the finite-stage closed relation denominator. This is the descent criterion needed to factor the algebraic map through the separated completed differential module.
Show proof
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
crossedDifferentialModuleLiftLinear
(R := ZCCompletedGroupAlgebra C H)
(zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
C hC hForm psi hpsi hfopen) x = 0 := by
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
letI : TopologicalSpace
(CrossedDifferentialPreModule (ZCCompletedGroupAlgebra C H) G) :=
zcCompletedDifferentialPreModuleNaturalTopology C psi.toMonoidHom
letI : T1Space
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
t1Space_kernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen
have hxcl :
x ∈ closure
((crossedDifferentialRelationSubmodule
(zcCompletedGroupAlgebraScalar C psi.toMonoidHom) :
Submodule (ZCCompletedGroupAlgebra C H)
(CrossedDifferentialPreModule (ZCCompletedGroupAlgebra C H) G)) :
Set (CrossedDifferentialPreModule (ZCCompletedGroupAlgebra C H) G)) := by
have hEq :=
closure_crossedDifferentialRelationSubmodule_eq_finiteClosedSubmodule
C psi.toMonoidHom hdir
rw [hEq]
exact hx
have hker_closed :
IsClosed
((fun y : CrossedDifferentialPreModule (ZCCompletedGroupAlgebra C H) G =>
crossedDifferentialModuleLiftLinear
(R := ZCCompletedGroupAlgebra C H)
(zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
C hC hForm psi hpsi hfopen) y) ⁻¹'
({0} : Set
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen))) :=
isClosed_singleton.preimage hcont
have hrel_subset_ker :
((crossedDifferentialRelationSubmodule
(zcCompletedGroupAlgebraScalar C psi.toMonoidHom) :
Submodule (ZCCompletedGroupAlgebra C H)
(CrossedDifferentialPreModule (ZCCompletedGroupAlgebra C H) G)) :
Set (CrossedDifferentialPreModule (ZCCompletedGroupAlgebra C H) G)) ⊆
((fun y : CrossedDifferentialPreModule (ZCCompletedGroupAlgebra C H) G =>
crossedDifferentialModuleLiftLinear
(R := ZCCompletedGroupAlgebra C H)
(zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
C hC hForm psi hpsi hfopen) y) ⁻¹'
({0} : Set
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen))) := by
intro y hy
exact
(crossedDifferentialRelationSubmodule_le_ker
(A := KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)
(zcCompletedGroupAlgebraScalar C psi.toMonoidHom)
(zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
C hC hForm psi hpsi hfopen)
(zcCompletedGASourceBoundaryToKerAugClosedQuot_isTargetCompletedCrossedDiff_of_surj
C hC hForm psi hpsi hfopen)) hy
exact closure_minimal hrel_subset_ker hker_closed hxclProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem zcDiffToKerAugClosedQuotOfSurj_kills_finiteClosedSubmodule_of_continuous_lift_of_surj
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(hcont :
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
@Continuous
(CrossedDifferentialPreModule (ZCCompletedGroupAlgebra C H) G)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)
(zcCompletedDifferentialPreModuleNaturalTopology C psi.toMonoidHom)
inferInstance
(crossedDifferentialModuleLiftLinear
(R := ZCCompletedGroupAlgebra C H)
(zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
C hC hForm psi hpsi hfopen)))
{x : CrossedDifferentialPreModule (ZCCompletedGroupAlgebra C H) G}
(hx : x ∈ zcCompletedDifferentialRelationFiniteClosedSubmodule C psi.toMonoidHom) :
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)Version of zcDiffToKerAugClosedQuotOfSurj_kills_finiteClosedSubmodule_of_continuous_lift with nonemptiness and directedness of finite stages supplied by the continuous source map.
Show proof
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
crossedDifferentialModuleLiftLinear
(R := ZCCompletedGroupAlgebra C H)
(zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
C hC hForm psi hpsi hfopen) x = 0 := by
letI : Nonempty (ZCCompletedDifferentialModuleIndex C psi.toMonoidHom) :=
nonempty_zcCompletedDifferentialModuleIndex C hC psi
exact
zcDiffToKerAugClosedQuotOfSurj_kills_finiteClosedSubmodule_of_continuous_lift
C hC hForm psi hpsi hfopen
(directed_zcCompletedDifferentialModuleIndex C hForm hC psi)
hcont hxProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□def zcSepDiffToKerAugClosedQuotOfSurjective
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
[Nonempty (ZCCompletedDifferentialModuleIndex C psi.toMonoidHom)]
(hdir : Directed (· ≤ ·)
(id : ZCCompletedDifferentialModuleIndex C psi.toMonoidHom →
ZCCompletedDifferentialModuleIndex C psi.toMonoidHom))
(hcont :
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
@Continuous
(CrossedDifferentialPreModule (ZCCompletedGroupAlgebra C H) G)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)
(zcCompletedDifferentialPreModuleNaturalTopology C psi.toMonoidHom)
inferInstance
(crossedDifferentialModuleLiftLinear
(R := ZCCompletedGroupAlgebra C H)
(zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
C hC hForm psi hpsi hfopen))) :
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom →ₗ[ZCCompletedGroupAlgebra C H]
KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen := by
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
exact
(zcCompletedDifferentialRelationFiniteClosedSubmodule C psi.toMonoidHom).liftQ
(crossedDifferentialModuleLiftLinear
(R := ZCCompletedGroupAlgebra C H)
(zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
C hC hForm psi hpsi hfopen))
(by
intro x hx
rw [LinearMap.mem_ker]
exact
zcDiffToKerAugClosedQuotOfSurj_kills_finiteClosedSubmodule_of_continuous_lift
C hC hForm psi hpsi hfopen hdir hcont hx)Under the explicit continuity hypothesis for the pre-quotient source-boundary lift, the closed source augmentation quotient receives the separated completed universal differential module.
def zcSepDiffToKerAugClosedQuotOfSurjectiveOfContinuousLift
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(hcont :
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
@Continuous
(CrossedDifferentialPreModule (ZCCompletedGroupAlgebra C H) G)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)
(zcCompletedDifferentialPreModuleNaturalTopology C psi.toMonoidHom)
inferInstance
(crossedDifferentialModuleLiftLinear
(R := ZCCompletedGroupAlgebra C H)
(zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
C hC hForm psi hpsi hfopen))) :
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom →ₗ[ZCCompletedGroupAlgebra C H]
KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen := by
letI : Nonempty (ZCCompletedDifferentialModuleIndex C psi.toMonoidHom) :=
nonempty_zcCompletedDifferentialModuleIndex C hC psi
exact
zcSepDiffToKerAugClosedQuotOfSurjective
C hC hForm psi hpsi hfopen
(directed_zcCompletedDifferentialModuleIndex C hForm hC psi)
hcont
@[simp 900]Public version of the closed-augmentation descent map with finite-stage nonemptiness and directedness supplied by the continuous source map.
theorem zcSepDiffToKerAugClosedQuotOfSurjective_universal
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
[Nonempty (ZCCompletedDifferentialModuleIndex C psi.toMonoidHom)]
(hdir : Directed (· ≤ ·)
(id : ZCCompletedDifferentialModuleIndex C psi.toMonoidHom →
ZCCompletedDifferentialModuleIndex C psi.toMonoidHom))
(hcont :
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
@Continuous
(CrossedDifferentialPreModule (ZCCompletedGroupAlgebra C H) G)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)
(zcCompletedDifferentialPreModuleNaturalTopology C psi.toMonoidHom)
inferInstance
(crossedDifferentialModuleLiftLinear
(R := ZCCompletedGroupAlgebra C H)
(zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
C hC hForm psi hpsi hfopen)))
(g : G) :
zcSepDiffToKerAugClosedQuotOfSurjective
C hC hForm psi hpsi hfopen hdir hcont
(zcSeparatedUniversalDifferential C psi.toMonoidHom g) =
zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
C hC hForm psi hpsi hfopen gThe separated differential module maps universally to the closed kernel-augmentation quotient in the surjective case.
Show proof
by
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
rw [zcSepDiffToKerAugClosedQuotOfSurjective,
zcSeparatedUniversalDifferential, Submodule.mkQ_apply, Submodule.liftQ_apply]
simp only [crossedDifferentialModuleLiftLinear_single, one_smul]
@[simp 900]Proof. Unfold the relevant Fox differential, quotient, or finite-stage map. The asserted formula is checked on group-ring generators or quotient classes; linearity, the crossed-derivation rule, and the defining augmentation/kernel relations extend the calculation to the whole module.
□theorem zcSepDiffToKerAugClosedQuotOfSurjectiveOfContinuousLift_universal
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(hcont :
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
@Continuous
(CrossedDifferentialPreModule (ZCCompletedGroupAlgebra C H) G)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)
(zcCompletedDifferentialPreModuleNaturalTopology C psi.toMonoidHom)
inferInstance
(crossedDifferentialModuleLiftLinear
(R := ZCCompletedGroupAlgebra C H)
(zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
C hC hForm psi hpsi hfopen)))
(g : G) :
zcSepDiffToKerAugClosedQuotOfSurjectiveOfContinuousLift
C hC hForm psi hpsi hfopen hcont
(zcSeparatedUniversalDifferential C psi.toMonoidHom g) =
zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
C hC hForm psi hpsi hfopen gThe lifted map from the separated completed differential module to the closed augmentation-kernel quotient sends each separated universal differential to the corresponding boundary quotient.
Show proof
by
letI : Nonempty (ZCCompletedDifferentialModuleIndex C psi.toMonoidHom) :=
nonempty_zcCompletedDifferentialModuleIndex C hC psi
exact
zcSepDiffToKerAugClosedQuotOfSurjective_universal
C hC hForm psi hpsi hfopen
(directed_zcCompletedDifferentialModuleIndex C hForm hC psi)
hcont gProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem zcSepDiffToKerAugClosedQuotOfSurjective_comp_toSep
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
[Nonempty (ZCCompletedDifferentialModuleIndex C psi.toMonoidHom)]
(hdir : Directed (· ≤ ·)
(id : ZCCompletedDifferentialModuleIndex C psi.toMonoidHom →
ZCCompletedDifferentialModuleIndex C psi.toMonoidHom))
(hcont :
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
@Continuous
(CrossedDifferentialPreModule (ZCCompletedGroupAlgebra C H) G)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)
(zcCompletedDifferentialPreModuleNaturalTopology C psi.toMonoidHom)
inferInstance
(crossedDifferentialModuleLiftLinear
(R := ZCCompletedGroupAlgebra C H)
(zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
C hC hForm psi hpsi hfopen))) :
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)The separated closed-augmentation map is the factorization of the algebraic closed-augmentation map through \(A_{\psi}(C) \to A_{\psi}(C)_{\mathrm{sep}}\).
Show proof
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
(zcSepDiffToKerAugClosedQuotOfSurjective
C hC hForm psi hpsi hfopen hdir hcont).comp
(zcCompletedDifferentialModuleToSeparated C psi.toMonoidHom) =
zcCompletedDifferentialModuleToKernelAugmentationClosedQuotientOfSurjective
C hC hForm psi hpsi hfopen := by
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
apply zcCompletedDifferentialModuleHom_ext C psi.toMonoidHom
intro g
rw [LinearMap.comp_apply,
zcCompletedDifferentialModuleToSeparated_universal,
zcSepDiffToKerAugClosedQuotOfSurjective_universal,
zcDiffToKerAugClosedQuotOfSurj_universal]Proof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem zcSepDiffToKerAugClosedQuotOfSurjectiveOfContinuousLift_comp_toSep
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(hcont :
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
@Continuous
(CrossedDifferentialPreModule (ZCCompletedGroupAlgebra C H) G)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)
(zcCompletedDifferentialPreModuleNaturalTopology C psi.toMonoidHom)
inferInstance
(crossedDifferentialModuleLiftLinear
(R := ZCCompletedGroupAlgebra C H)
(zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
C hC hForm psi hpsi hfopen))) :
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)The continuous-lift map to the closed kernel-augmentation quotient composes with separation as expected.
Show proof
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
(zcSepDiffToKerAugClosedQuotOfSurjectiveOfContinuousLift
C hC hForm psi hpsi hfopen hcont).comp
(zcCompletedDifferentialModuleToSeparated C psi.toMonoidHom) =
zcCompletedDifferentialModuleToKernelAugmentationClosedQuotientOfSurjective
C hC hForm psi hpsi hfopen := by
letI : Nonempty (ZCCompletedDifferentialModuleIndex C psi.toMonoidHom) :=
nonempty_zcCompletedDifferentialModuleIndex C hC psi
exact
zcSepDiffToKerAugClosedQuotOfSurjective_comp_toSep
C hC hForm psi hpsi hfopen
(directed_zcCompletedDifferentialModuleIndex C hForm hC psi)
hcontProof. Unfold the relevant Fox differential, quotient, or finite-stage map. The asserted formula is checked on group-ring generators or quotient classes; linearity, the crossed-derivation rule, and the defining augmentation/kernel relations extend the calculation to the whole module.
□theorem continuous_zcSepDiffToKerAugClosedQuotOfSurjective
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
[Nonempty (ZCCompletedDifferentialModuleIndex C psi.toMonoidHom)]
(hdir : Directed (· ≤ ·)
(id : ZCCompletedDifferentialModuleIndex C psi.toMonoidHom →
ZCCompletedDifferentialModuleIndex C psi.toMonoidHom))
(hcont :
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
@Continuous
(CrossedDifferentialPreModule (ZCCompletedGroupAlgebra C H) G)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)
(zcCompletedDifferentialPreModuleNaturalTopology C psi.toMonoidHom)
inferInstance
(crossedDifferentialModuleLiftLinear
(R := ZCCompletedGroupAlgebra C H)
(zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
C hC hForm psi hpsi hfopen))) :
letI : TopologicalSpace (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom)The descended forward map to the closed augmentation quotient is continuous once the pre-quotient source-boundary lift is continuous.
Show proof
zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
@Continuous
(ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)
(zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom)
inferInstance
(zcSepDiffToKerAugClosedQuotOfSurjective
C hC hForm psi hpsi hfopen hdir hcont) := by
letI : TopologicalSpace (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
rw [continuous_zcSeparatedCompletedDifferentialModule_iff_comp_mkQ
(C := C) (G := G) (H := H) (ψ := psi.toMonoidHom)]
have hcomp :
(fun x : CrossedDifferentialPreModule (ZCCompletedGroupAlgebra C H) G =>
zcSepDiffToKerAugClosedQuotOfSurjective
C hC hForm psi hpsi hfopen hdir hcont
((zcCompletedDifferentialRelationFiniteClosedSubmodule C psi.toMonoidHom).mkQ x)) =
crossedDifferentialModuleLiftLinear
(R := ZCCompletedGroupAlgebra C H)
(zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
C hC hForm psi hpsi hfopen) := by
funext x
rw [zcSepDiffToKerAugClosedQuotOfSurjective,
Submodule.mkQ_apply, Submodule.liftQ_apply]
simpa [hcomp] using hcontProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem continuous_zcSepDiffToKerAugClosedQuotOfSurjectiveOfContinuousLift
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(hcont :
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
@Continuous
(CrossedDifferentialPreModule (ZCCompletedGroupAlgebra C H) G)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)
(zcCompletedDifferentialPreModuleNaturalTopology C psi.toMonoidHom)
inferInstance
(crossedDifferentialModuleLiftLinear
(R := ZCCompletedGroupAlgebra C H)
(zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
C hC hForm psi hpsi hfopen))) :
letI : TopologicalSpace (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom)Continuity of the forward map when the finite-stage index data are supplied by the continuous source map.
Show proof
zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
@Continuous
(ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)
(zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom)
inferInstance
(zcSepDiffToKerAugClosedQuotOfSurjectiveOfContinuousLift
C hC hForm psi hpsi hfopen hcont) := by
letI : Nonempty (ZCCompletedDifferentialModuleIndex C psi.toMonoidHom) :=
nonempty_zcCompletedDifferentialModuleIndex C hC psi
exact
continuous_zcSepDiffToKerAugClosedQuotOfSurjective
C hC hForm psi hpsi hfopen
(directed_zcCompletedDifferentialModuleIndex C hForm hC psi)
hcontProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem zcDiffToKerAugClosedQuotOfSurj_surj
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi) :
Function.Surjective
(zcCompletedDifferentialModuleToKernelAugmentationClosedQuotientOfSurjective
C hC hForm psi hpsi hfopen)The map from the completed differential module to the closed augmentation quotient is surjective when finite-stage representatives can be lifted.
Show proof
by
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
intro x
refine Submodule.Quotient.induction_on
(p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen) x ?_
intro y
let L :=
zcCompletedDifferentialModuleToKernelAugmentationClosedQuotientOfSurjective
C hC hForm psi hpsi hfopen
let P : zcCompletedGroupAlgebraStandardAugmentationIdeal C G → Prop := fun y =>
∃ m : ZCCompletedDifferentialModule C psi.toMonoidHom,
L m = Submodule.Quotient.mk
(p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen) y
have hy : P y := by
have hyspan : (y : ZCCompletedGroupAlgebra C G) ∈
Submodule.span (ZCCompletedGroupAlgebra C G)
(Set.range fun h : G => zcGroupLike C G h - 1) := by
change (y : ZCCompletedGroupAlgebra C G) ∈
zcCompletedGroupAlgebraStandardAugmentationIdeal C G
exact y.2
refine Submodule.span_induction
(p := fun z hz =>
P
⟨z, by
simpa [zcCompletedGroupAlgebraStandardAugmentationIdeal_eq_span] using hz⟩)
?hgen ?hzero ?hadd ?hsmul hyspan
· rintro _ ⟨g, rfl⟩
refine ⟨zcUniversalDifferential C psi.toMonoidHom g, ?_⟩
rw [zcDiffToKerAugClosedQuotOfSurj_universal]
rfl
· refine ⟨0, ?_⟩
change (0 : KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) =
Submodule.Quotient.mk
(p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen)
(0 : zcCompletedGroupAlgebraStandardAugmentationIdeal C G)
rw [Submodule.Quotient.mk_zero]
· intro a b ha hb hpa hpb
rcases hpa with ⟨ma, hma⟩
rcases hpb with ⟨mb, hmb⟩
refine ⟨ma + mb, ?_⟩
rw [map_add, hma, hmb, ← Submodule.Quotient.mk_add]
apply congrArg (fun t : zcCompletedGroupAlgebraStandardAugmentationIdeal C G =>
Submodule.Quotient.mk
(p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen) t)
exact Subtype.ext rfl
· intro a b hb hpb
rcases hpb with ⟨m, hm⟩
refine ⟨zcCompletedGroupAlgebraMap C hC psi a • m, ?_⟩
rw [map_smul, hm]
calc
zcCompletedGroupAlgebraMap C hC psi a •
Submodule.Quotient.mk
(p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen)
⟨b, by
simpa [zcCompletedGroupAlgebraStandardAugmentationIdeal_eq_span] using hb⟩ =
a •
Submodule.Quotient.mk
(p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen)
⟨b, by
simpa [zcCompletedGroupAlgebraStandardAugmentationIdeal_eq_span] using hb⟩ := by
exact
kerAugClosedQuotTargetCompletedModuleOfSurj_map_smul
C hC hForm psi hpsi hfopen a
(Submodule.Quotient.mk
(p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen)
⟨b, by
simpa [zcCompletedGroupAlgebraStandardAugmentationIdeal_eq_span] using hb⟩)
_ = Submodule.Quotient.mk
(p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen)
⟨a • b, by
rw [zcCompletedGroupAlgebraStandardAugmentationIdeal_eq_span]
exact Submodule.smul_mem
(Submodule.span (ZCCompletedGroupAlgebra C G)
(Set.range fun h : G => zcGroupLike C G h - 1)) a hb⟩ := by
rw [← Submodule.Quotient.mk_smul]
apply congrArg (fun t : zcCompletedGroupAlgebraStandardAugmentationIdeal C G =>
Submodule.Quotient.mk
(p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen) t)
exact Subtype.ext rfl
exact hyProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem zcSepDiffToKerAugClosedQuotOfSurjective_surj
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
[Nonempty (ZCCompletedDifferentialModuleIndex C psi.toMonoidHom)]
(hdir : Directed (· ≤ ·)
(id : ZCCompletedDifferentialModuleIndex C psi.toMonoidHom →
ZCCompletedDifferentialModuleIndex C psi.toMonoidHom))
(hcont :
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
@Continuous
(CrossedDifferentialPreModule (ZCCompletedGroupAlgebra C H) G)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)
(zcCompletedDifferentialPreModuleNaturalTopology C psi.toMonoidHom)
inferInstance
(crossedDifferentialModuleLiftLinear
(R := ZCCompletedGroupAlgebra C H)
(zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
C hC hForm psi hpsi hfopen))) :
Function.Surjective
(zcSepDiffToKerAugClosedQuotOfSurjective
C hC hForm psi hpsi hfopen hdir hcont)The separated closed-augmentation map is surjective under the same pre-quotient continuity hypothesis needed for the descent.
Show proof
by
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
intro y
rcases
zcDiffToKerAugClosedQuotOfSurj_surj
C hC hForm psi hpsi hfopen y with
⟨m, hm⟩
refine ⟨zcCompletedDifferentialModuleToSeparated C psi.toMonoidHom m, ?_⟩
have hfactor :=
congrArg (fun L : ZCCompletedDifferentialModule C psi.toMonoidHom →ₗ[ZCCompletedGroupAlgebra C H]
KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen => L m)
(zcSepDiffToKerAugClosedQuotOfSurjective_comp_toSep
C hC hForm psi hpsi hfopen hdir hcont)
simpa [LinearMap.comp_apply] using hfactor.trans hmProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem zcSepDiffToKerAugClosedQuotOfSurjectiveOfContinuousLift_surj
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(hcont :
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
@Continuous
(CrossedDifferentialPreModule (ZCCompletedGroupAlgebra C H) G)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)
(zcCompletedDifferentialPreModuleNaturalTopology C psi.toMonoidHom)
inferInstance
(crossedDifferentialModuleLiftLinear
(R := ZCCompletedGroupAlgebra C H)
(zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
C hC hForm psi hpsi hfopen))) :
Function.Surjective
(zcSepDiffToKerAugClosedQuotOfSurjectiveOfContinuousLift
C hC hForm psi hpsi hfopen hcont)The continuous-lift map to the closed kernel-augmentation quotient is surjective.
Show proof
by
letI : Nonempty (ZCCompletedDifferentialModuleIndex C psi.toMonoidHom) :=
nonempty_zcCompletedDifferentialModuleIndex C hC psi
exact
zcSepDiffToKerAugClosedQuotOfSurjective_surj
C hC hForm psi hpsi hfopen
(directed_zcCompletedDifferentialModuleIndex C hForm hC psi)
hcontProof. Unfold the relevant Fox differential, quotient, or finite-stage map. The asserted formula is checked on group-ring generators or quotient classes; linearity, the crossed-derivation rule, and the defining augmentation/kernel relations extend the calculation to the whole module.
□def zcSeparatedCompletedDifferentialModuleToKernelAugmentationClosedQuotient
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi) :
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom →ₗ[ZCCompletedGroupAlgebra C H]
KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen := by
let hcont :=
continuous_crossedDiffModuleLiftLinear_sourceBoundaryToKerAugClosedQuot
C hC hForm psi hpsi hfopen
exact
zcSepDiffToKerAugClosedQuotOfSurjectiveOfContinuousLift
C hC hForm psi hpsi hfopen hcont
@[simp]The separated completed universal differential module maps to the closed source augmentation quotient by \(dg \mapsto\) \([g]-1\). The pre-quotient lift continuity is supplied by the finite-stage factorization theorem.
theorem zcSepDiffToKerAugClosedQuot_universal
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi) (g : G) :
zcSeparatedCompletedDifferentialModuleToKernelAugmentationClosedQuotient
C hC hForm psi hpsi hfopen
(zcSeparatedUniversalDifferential C psi.toMonoidHom g) =
zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
C hC hForm psi hpsi hfopen gThe separated closed augmentation-quotient map sends the separated universal differential to the closed augmentation quotient class.
Show proof
by
let hcont :=
continuous_crossedDiffModuleLiftLinear_sourceBoundaryToKerAugClosedQuot
C hC hForm psi hpsi hfopen
simpa [zcSeparatedCompletedDifferentialModuleToKernelAugmentationClosedQuotient, hcont] using
zcSepDiffToKerAugClosedQuotOfSurjectiveOfContinuousLift_universal
C hC hForm psi hpsi hfopen hcont gProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem zcSepDiffToKerAugClosedQuot_comp_toSep
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi) :
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)The unconditional separated closed-augmentation map factors the algebraic map through \(A_{\psi}(C) \to A_{\psi}(C)_{\mathrm{sep}}\).
Show proof
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
(zcSeparatedCompletedDifferentialModuleToKernelAugmentationClosedQuotient
C hC hForm psi hpsi hfopen).comp
(zcCompletedDifferentialModuleToSeparated C psi.toMonoidHom) =
zcCompletedDifferentialModuleToKernelAugmentationClosedQuotientOfSurjective
C hC hForm psi hpsi hfopen := by
let hcont :=
continuous_crossedDiffModuleLiftLinear_sourceBoundaryToKerAugClosedQuot
C hC hForm psi hpsi hfopen
simpa [zcSeparatedCompletedDifferentialModuleToKernelAugmentationClosedQuotient, hcont] using
zcSepDiffToKerAugClosedQuotOfSurjectiveOfContinuousLift_comp_toSep
C hC hForm psi hpsi hfopen hcontProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem continuous_zcSepDiffToKerAugClosedQuot
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi) :
letI : TopologicalSpace (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom)The separated closed-augmentation map is continuous, with the pre-quotient lift continuity provided by the finite-stage factorization theorem.
Show proof
zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
@Continuous
(ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)
(zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom)
inferInstance
(zcSeparatedCompletedDifferentialModuleToKernelAugmentationClosedQuotient
C hC hForm psi hpsi hfopen) := by
let hcont :=
continuous_crossedDiffModuleLiftLinear_sourceBoundaryToKerAugClosedQuot
C hC hForm psi hpsi hfopen
simpa [zcSeparatedCompletedDifferentialModuleToKernelAugmentationClosedQuotient, hcont] using
continuous_zcSepDiffToKerAugClosedQuotOfSurjectiveOfContinuousLift
C hC hForm psi hpsi hfopen hcontProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem zcSepDiffToKerAugClosedQuot_surj
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi) :
Function.Surjective
(zcSeparatedCompletedDifferentialModuleToKernelAugmentationClosedQuotient
C hC hForm psi hpsi hfopen)The separated map to the closed kernel-augmentation quotient is surjective.
Show proof
by
let hcont :=
continuous_crossedDiffModuleLiftLinear_sourceBoundaryToKerAugClosedQuot
C hC hForm psi hpsi hfopen
simpa [zcSeparatedCompletedDifferentialModuleToKernelAugmentationClosedQuotient, hcont] using
zcSepDiffToKerAugClosedQuotOfSurjectiveOfContinuousLift_surj
C hC hForm psi hpsi hfopen hcont
@[simp]Proof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem kerAugIdealQuotToZCSepDiffLinear_mk
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G) :
kerAugIdealQuotToZCSepDiffLinear
C hC hForm psi hpsi hfopen
(Submodule.Quotient.mk
(p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen) x) =
stdAugIdealToZCSepDiff
C hC psi xThe reverse map from the closed augmentation quotient to the separated differential module is evaluated on quotient classes by the constructed linear representative.
Show proof
by
simp only [ContinuousMonoidHom.coe_toMonoidHom, kerAugIdealQuotToZCSepDiffLinear,
kerAugIdealQuotToZCSepDiffLinearOfContStdMap, kerAugIdealQuotToZCSepDiffLinearOfClosedKill,
kerAugIdealQuotToZCSepDiffLinearOfBoundaryKernelOfClosedKill,
kerAugIdealQuotToZCSepDiffOfBoundaryKernelOfClosedKill, LinearMap.coe_mk, AddHom.coe_mk, Submodule.liftQ_apply,
stdAugIdealToZCSepDiff]Proof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem kerAugIdealQuotToZCSepDiffLinear_comp_zcSepDiffToKerAugClosedQuot
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi) :
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)Composing the reverse closed-augmentation map with the forward separated map gives the identity.
Show proof
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
(kerAugIdealQuotToZCSepDiffLinear
C hC hForm psi hpsi hfopen).comp
(zcSeparatedCompletedDifferentialModuleToKernelAugmentationClosedQuotient
C hC hForm psi hpsi hfopen) =
LinearMap.id := by
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
apply zcSeparatedCompletedDifferentialModuleHom_ext C psi.toMonoidHom
intro g
rw [LinearMap.comp_apply,
zcSepDiffToKerAugClosedQuot_universal]
simpa [zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient] using
kerAugIdealQuotToZCSepDiffLinear_boundary
C hC hForm psi hpsi hfopen gProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem zcSepDiffToKerAugClosedQuot_comp_kerAugIdealQuotToZCSepDiffLinear
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi) :
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)Composing the forward separated map with the reverse closed-augmentation map gives the identity.
Show proof
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
(zcSeparatedCompletedDifferentialModuleToKernelAugmentationClosedQuotient
C hC hForm psi hpsi hfopen).comp
(kerAugIdealQuotToZCSepDiffLinear
C hC hForm psi hpsi hfopen) =
LinearMap.id := by
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
letI : Module (ZCCompletedGroupAlgebra C G)
(ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
apply LinearMap.ext
intro x
refine Submodule.Quotient.induction_on
(p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen) x ?_
intro y
let F :=
zcSeparatedCompletedDifferentialModuleToKernelAugmentationClosedQuotient
C hC hForm psi hpsi hfopen
let S :=
stdAugIdealToZCSepDiff
C hC psi
let Q : zcCompletedGroupAlgebraStandardAugmentationIdeal C G →
KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen :=
fun y => Submodule.Quotient.mk
(p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen) y
have hmk :
kerAugIdealQuotToZCSepDiffLinear
C hC hForm psi hpsi hfopen (Q y) =
S y := by
exact
kerAugIdealQuotToZCSepDiffLinear_mk
C hC hForm psi hpsi hfopen y
change F
(kerAugIdealQuotToZCSepDiffLinear
C hC hForm psi hpsi hfopen (Q y)) = Q y
rw [hmk]
let P : zcCompletedGroupAlgebraStandardAugmentationIdeal C G → Prop := fun y =>
F (S y) = Q y
have hyspan : (y : ZCCompletedGroupAlgebra C G) ∈
Submodule.span (ZCCompletedGroupAlgebra C G)
(Set.range fun h : G => zcGroupLike C G h - 1) := by
change (y : ZCCompletedGroupAlgebra C G) ∈
zcCompletedGroupAlgebraStandardAugmentationIdeal C G
exact y.2
exact
(Submodule.span_induction
(p := fun z hz =>
∀ y' : zcCompletedGroupAlgebraStandardAugmentationIdeal C G,
(y' : ZCCompletedGroupAlgebra C G) = z → P y')
(by
rintro _ ⟨g, rfl⟩ y' hy'
have hy'' :
y' =
⟨zcCompletedGroupAlgebraBoundary C (MonoidHom.id G) g,
zcCompletedGroupAlgebraBoundary_mem_standardAugmentationIdeal
C G (MonoidHom.id G) g⟩ := by
apply Subtype.ext
simpa [zcCompletedGroupAlgebraBoundary] using hy'
rw [hy'']
let yg : zcCompletedGroupAlgebraStandardAugmentationIdeal C G :=
⟨zcCompletedGroupAlgebraBoundary C (MonoidHom.id G) g,
zcCompletedGroupAlgebraBoundary_mem_standardAugmentationIdeal
C G (MonoidHom.id G) g⟩
change P yg
have hSg :
S yg = zcSeparatedUniversalDifferential C psi.toMonoidHom g := by
exact
stdAugIdealToZCSepDiff_boundary
C hC psi g
dsimp [P, Q]
calc
F (S yg) = F (zcSeparatedUniversalDifferential C psi.toMonoidHom g) := by
exact congrArg F hSg
_ =
zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
C hC hForm psi hpsi hfopen g := by
rw [zcSepDiffToKerAugClosedQuot_universal]
_ = Submodule.Quotient.mk yg := rfl)
(by
intro y' hy'
have hy'' : y' = 0 := by
apply Subtype.ext
simpa using hy'
rw [hy'']
change P (0 : zcCompletedGroupAlgebraStandardAugmentationIdeal C G)
simp only [ContinuousMonoidHom.coe_toMonoidHom, map_zero, Submodule.Quotient.mk_zero, P, Q])
(by
intro a b ha hb hpa hpb y' hy'
let ya : zcCompletedGroupAlgebraStandardAugmentationIdeal C G :=
⟨a, by
simpa [zcCompletedGroupAlgebraStandardAugmentationIdeal_eq_span] using ha⟩
let yb : zcCompletedGroupAlgebraStandardAugmentationIdeal C G :=
⟨b, by
simpa [zcCompletedGroupAlgebraStandardAugmentationIdeal_eq_span] using hb⟩
have hpa' : P ya := hpa ya rfl
have hpb' : P yb := hpb yb rfl
change F (S ya) = Q ya at hpa'
change F (S yb) = Q yb at hpb'
have hy'' : y' = ya + yb := by
apply Subtype.ext
simpa [ya, yb] using hy'
rw [hy'']
change P (ya + yb)
change F (S (ya + yb)) = Q (ya + yb)
rw [map_add, map_add, hpa', hpb']
simp only [Submodule.Quotient.mk_add, Q])
(by
intro a b hb hpb y' hy'
let yb : zcCompletedGroupAlgebraStandardAugmentationIdeal C G :=
⟨b, by
simpa [zcCompletedGroupAlgebraStandardAugmentationIdeal_eq_span] using hb⟩
have hpb' : P yb := hpb yb rfl
change F (S yb) = Q yb at hpb'
have hy'' : y' = a • yb := by
apply Subtype.ext
simpa [yb] using hy'
rw [hy'']
change P (a • yb)
change F (S (a • yb)) = Q (a • yb)
have hsource :
a • S yb =
zcCompletedGroupAlgebraMap C hC psi a • S yb := by
exact
(zcSeparatedCompletedDifferentialModule_source_map_smul
C hC psi a (S yb)).symm
calc
F (S (a • yb)) = F (a • S yb) := by
rw [map_smul]
_ = F (zcCompletedGroupAlgebraMap C hC psi a • S yb) := by
rw [hsource]
_ = zcCompletedGroupAlgebraMap C hC psi a • F (S yb) := by
rw [map_smul]
_ = zcCompletedGroupAlgebraMap C hC psi a • Q yb := by
rw [hpb']
_ = a • Q yb := by
exact
kerAugClosedQuotTargetCompletedModuleOfSurj_map_smul
C hC hForm psi hpsi hfopen a (Q yb)
_ = Q (a • yb) := by
dsimp [Q])
hyspan) y rflProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□def zcSepDiffEquivKerAugClosedQuot_of_surj
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi) :
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom
≃ₗ[ZCCompletedGroupAlgebra C H]
KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen := by
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
exact
LinearEquiv.ofLinear
(zcSeparatedCompletedDifferentialModuleToKernelAugmentationClosedQuotient
C hC hForm psi hpsi hfopen)
(kerAugIdealQuotToZCSepDiffLinear
C hC hForm psi hpsi hfopen)
(zcSepDiffToKerAugClosedQuot_comp_kerAugIdealQuotToZCSepDiffLinear
C hC hForm psi hpsi hfopen)
(kerAugIdealQuotToZCSepDiffLinear_comp_zcSepDiffToKerAugClosedQuot
C hC hForm psi hpsi hfopen)
@[simp]The separated completed differential module is the closed source augmentation quotient for a surjective open continuous homomorphism.
theorem zcSepDiffEquivKerAugClosedQuot_of_surj_apply
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(x : ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :
zcSepDiffEquivKerAugClosedQuot_of_surj
C hC hForm psi hpsi hfopen x =
zcSeparatedCompletedDifferentialModuleToKernelAugmentationClosedQuotient
C hC hForm psi hpsi hfopen xThe Fox-coordinate equivalence is evaluated by the finite-stage coordinate formula in the completed differential complex.
Show proof
rfl
@[simp]Proof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem zcSepDiffEquivKerAugClosedQuot_of_surj_symm_apply
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(x : KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :
(zcSepDiffEquivKerAugClosedQuot_of_surj
C hC hForm psi hpsi hfopen).symm x =
kerAugIdealQuotToZCSepDiffLinear
C hC hForm psi hpsi hfopen xThe inverse Fox-coordinate equivalence is evaluated by reconstructing the class from its completed coordinate data.
Show proof
rflProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□def zcApsiEquivKerAugClosedQuot_of_surj
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi) :
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
ZCApsi C psi.toMonoidHom ≃ₗ[ZCCompletedGroupAlgebra C H]
KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen :=
zcSepDiffEquivKerAugClosedQuot_of_surj
C hC hForm psi hpsi hfopen
@[simp]The closed source augmentation quotient equivalence identifies \(A_{\psi}(C)\) over \(\mathbb{Z}_C\) with the separated completed differential module, rather than with the raw algebraic quotient.
theorem zcApsiEquivKerAugClosedQuot_of_surj_apply
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(x : ZCApsi C psi.toMonoidHom) :
zcApsiEquivKerAugClosedQuot_of_surj
C hC hForm psi hpsi hfopen x =
zcSeparatedCompletedDifferentialModuleToKernelAugmentationClosedQuotient
C hC hForm psi hpsi hfopen xThe Fox-coordinate equivalence is evaluated by the finite-stage coordinate formula in the completed differential complex.
Show proof
rfl
@[simp]Proof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem zcApsiEquivKerAugClosedQuot_of_surj_symm_apply
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(x : KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :
(zcApsiEquivKerAugClosedQuot_of_surj
C hC hForm psi hpsi hfopen).symm x =
kerAugIdealQuotToZCSepDiffLinear
C hC hForm psi hpsi hfopen xThe inverse Fox-coordinate equivalence is evaluated by reconstructing the class from its completed coordinate data.
Show proof
rflProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem zcCompletedGAKerAugQuotTargetCompletedModuleOfSurjective_map_smul
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hker_mul :
∀ k : ZCCompletedGroupAlgebra C G,
k ∈ RingHom.ker (zcCompletedGroupAlgebraMap C hC psi) →
∀ y : zcCompletedGroupAlgebraStandardAugmentationIdeal C G,
(⟨k * (y : ZCCompletedGroupAlgebra C G),
(zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left k y.2⟩ :
zcCompletedGroupAlgebraStandardAugmentationIdeal C G) ∈
zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi)
(a : ZCCompletedGroupAlgebra C G) (x : KernelAugmentationIdealQuotient C psi) :
letI : Module (ZCCompletedGroupAlgebra C H) (KernelAugmentationIdealQuotient C psi)For a surjective target map, the completed target scalar action on the kernel-augmentation quotient is compatible with applying the induced algebra map.
Show proof
zcCompletedGAKerAugQuotTargetCompletedModuleOfSurjective_of_kernelMulStandard_le
C hC hForm psi hpsi hker_mul
zcCompletedGroupAlgebraMap C hC psi a • x = a • x := by
letI : Module (ZCCompletedGroupAlgebra C H) (KernelAugmentationIdealQuotient C psi) :=
zcCompletedGAKerAugQuotTargetCompletedModuleOfSurjective_of_kernelMulStandard_le
C hC hForm psi hpsi hker_mul
change zcCompletedGroupAlgebraTargetLiftOfSurjective C hC hForm psi hpsi
(zcCompletedGroupAlgebraMap C hC psi a) • x =
a • x
have hdiff :
zcCompletedGroupAlgebraTargetLiftOfSurjective C hC hForm psi hpsi
(zcCompletedGroupAlgebraMap C hC psi a) - a ∈
RingHom.ker (zcCompletedGroupAlgebraMap C hC psi) := by
change zcCompletedGroupAlgebraMap C hC psi
(zcCompletedGroupAlgebraTargetLiftOfSurjective C hC hForm psi hpsi
(zcCompletedGroupAlgebraMap C hC psi a) - a) = 0
rw [map_sub, zcCompletedGroupAlgebraMap_targetLiftOfSurjective, sub_self]
have hzero :=
zcCompletedGAKerAugQuot_ker_map_smul_eq_zero_of_kernelMulStandard_le
C hC psi hker_mul
(zcCompletedGroupAlgebraTargetLiftOfSurjective C hC hForm psi hpsi
(zcCompletedGroupAlgebraMap C hC psi a) - a) hdiff x
rw [sub_smul] at hzero
exact sub_eq_zero.mp hzeroProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem zcCompletedGAKerAugQuotTargetCompletedModuleOfSurjective_groupLike_smul
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hker_mul :
∀ k : ZCCompletedGroupAlgebra C G,
k ∈ RingHom.ker (zcCompletedGroupAlgebraMap C hC psi) →
∀ y : zcCompletedGroupAlgebraStandardAugmentationIdeal C G,
(⟨k * (y : ZCCompletedGroupAlgebra C G),
(zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left k y.2⟩ :
zcCompletedGroupAlgebraStandardAugmentationIdeal C G) ∈
zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi)
(g : G) (x : KernelAugmentationIdealQuotient C psi) :
letI : Module (ZCCompletedGroupAlgebra C H) (KernelAugmentationIdealQuotient C psi)Surjectivity of the induced map is obtained from surjectivity on the underlying quotient or dense algebraic model, together with closedness of the image in the completed target.
Show proof
zcCompletedGAKerAugQuotTargetCompletedModuleOfSurjective_of_kernelMulStandard_le
C hC hForm psi hpsi hker_mul
zcGroupLike C H (psi g) • x = zcGroupLike C G g • x := by
letI : Module (ZCCompletedGroupAlgebra C H) (KernelAugmentationIdealQuotient C psi) :=
zcCompletedGAKerAugQuotTargetCompletedModuleOfSurjective_of_kernelMulStandard_le
C hC hForm psi hpsi hker_mul
rw [← zcCompletedGroupAlgebraMap_groupLike (C := C) (hC := hC) psi g]
exact
zcCompletedGAKerAugQuotTargetCompletedModuleOfSurjective_map_smul
C hC hForm psi hpsi hker_mul (zcGroupLike C G g) xProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem sourceBoundaryToKerAug_isTargetCrossedDiff_of_surj_of_kernelMulStandard_le
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hker_mul :
∀ k : ZCCompletedGroupAlgebra C G,
k ∈ RingHom.ker (zcCompletedGroupAlgebraMap C hC psi) →
∀ y : zcCompletedGroupAlgebraStandardAugmentationIdeal C G,
(⟨k * (y : ZCCompletedGroupAlgebra C G),
(zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left k y.2⟩ :
zcCompletedGroupAlgebraStandardAugmentationIdeal C G) ∈
zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi) :
letI : Module (ZCCompletedGroupAlgebra C H) (KernelAugmentationIdealQuotient C psi)Under the explicit kernel-product hypothesis, the source boundary is a crossed differential for the descended completed target scalars.
Show proof
zcCompletedGAKerAugQuotTargetCompletedModuleOfSurjective_of_kernelMulStandard_le
C hC hForm psi hpsi hker_mul
IsCrossedDifferential
(zcCompletedGroupAlgebraScalar C psi.toMonoidHom)
(zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationQuotient C psi) := by
letI : Module (ZCCompletedGroupAlgebra C H) (KernelAugmentationIdealQuotient C psi) :=
zcCompletedGAKerAugQuotTargetCompletedModuleOfSurjective_of_kernelMulStandard_le
C hC hForm psi hpsi hker_mul
intro g h
have hsource :=
zcCompletedGASourceBoundaryToKerAugQuot_isCrossedDiff
C psi g h
rw [hsource]
congr 1
change zcGroupLike C G g •
zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationQuotient C psi h =
zcGroupLike C H (psi g) •
zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationQuotient C psi h
exact
(zcCompletedGAKerAugQuotTargetCompletedModuleOfSurjective_groupLike_smul
C hC hForm psi hpsi hker_mul g
(zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationQuotient C psi h)).symmProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□def zcDiffToKerAugQuotOfSurj_of_kernelMulStandard_le
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hker_mul :
∀ k : ZCCompletedGroupAlgebra C G,
k ∈ RingHom.ker (zcCompletedGroupAlgebraMap C hC psi) →
∀ y : zcCompletedGroupAlgebraStandardAugmentationIdeal C G,
(⟨k * (y : ZCCompletedGroupAlgebra C G),
(zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left k y.2⟩ :
zcCompletedGroupAlgebraStandardAugmentationIdeal C G) ∈
zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi) :
letI : Module (ZCCompletedGroupAlgebra C H) (KernelAugmentationIdealQuotient C psi) :=
zcCompletedGAKerAugQuotTargetCompletedModuleOfSurjective_of_kernelMulStandard_le
C hC hForm psi hpsi hker_mul
ZCCompletedDifferentialModule C psi.toMonoidHom →ₗ[ZCCompletedGroupAlgebra C H]
KernelAugmentationIdealQuotient C psi := by
letI : Module (ZCCompletedGroupAlgebra C H) (KernelAugmentationIdealQuotient C psi) :=
zcCompletedGAKerAugQuotTargetCompletedModuleOfSurjective_of_kernelMulStandard_le
C hC hForm psi hpsi hker_mul
exact
zcCompletedDifferentialModuleLift
(A := KernelAugmentationIdealQuotient C psi)
C psi.toMonoidHom
(zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationQuotient C psi)
(sourceBoundaryToKerAug_isTargetCrossedDiff_of_surj_of_kernelMulStandard_le
C hC hForm psi hpsi hker_mul)
@[simp 900]Under the explicit kernel-product hypothesis, the completed universal differential module maps to the algebraic source augmentation quotient by \(dg \mapsto [g]-1\). The hypothesis is the condition needed for the algebraic quotient \(I(G) / I(\ker \psi)I(G)\) to carry the completed target \(\mathbb{Z}_C\llbracket H\rrbracket\)-module structure.
theorem zcDiffToKerAugQuotOfSurj_of_kernelMulStandard_le_universal
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hker_mul :
∀ k : ZCCompletedGroupAlgebra C G,
k ∈ RingHom.ker (zcCompletedGroupAlgebraMap C hC psi) →
∀ y : zcCompletedGroupAlgebraStandardAugmentationIdeal C G,
(⟨k * (y : ZCCompletedGroupAlgebra C G),
(zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left k y.2⟩ :
zcCompletedGroupAlgebraStandardAugmentationIdeal C G) ∈
zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi)
(g : G) :
zcDiffToKerAugQuotOfSurj_of_kernelMulStandard_le
C hC hForm psi hpsi hker_mul
(zcUniversalDifferential C psi.toMonoidHom g) =
zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationQuotient C psi gUnder the kernel-product hypothesis, the algebraic source augmentation quotient map sends the universal differential to the class of \([g]-1\).
Show proof
by
letI : Module (ZCCompletedGroupAlgebra C H) (KernelAugmentationIdealQuotient C psi) :=
zcCompletedGAKerAugQuotTargetCompletedModuleOfSurjective_of_kernelMulStandard_le
C hC hForm psi hpsi hker_mul
exact
zcCompletedDifferentialModuleLift_universal
(A := KernelAugmentationIdealQuotient C psi)
C psi.toMonoidHom
(zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationQuotient C psi)
(sourceBoundaryToKerAug_isTargetCrossedDiff_of_surj_of_kernelMulStandard_le
C hC hForm psi hpsi hker_mul) gProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem zcDiffToKerAugQuotOfSurj_of_kernelMulStandard_le_surj
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hker_mul :
∀ k : ZCCompletedGroupAlgebra C G,
k ∈ RingHom.ker (zcCompletedGroupAlgebraMap C hC psi) →
∀ y : zcCompletedGroupAlgebraStandardAugmentationIdeal C G,
(⟨k * (y : ZCCompletedGroupAlgebra C G),
(zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left k y.2⟩ :
zcCompletedGroupAlgebraStandardAugmentationIdeal C G) ∈
zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi) :
Function.Surjective
(zcDiffToKerAugQuotOfSurj_of_kernelMulStandard_le
C hC hForm psi hpsi hker_mul)The algebraic map \(A_{\psi}(C) \to I(G) / I(\ker \psi)I(G)\) is onto once the algebraic quotient has the completed target scalar action supplied by the explicit kernel-product hypothesis.
Show proof
by
letI : Module (ZCCompletedGroupAlgebra C H) (KernelAugmentationIdealQuotient C psi) :=
zcCompletedGAKerAugQuotTargetCompletedModuleOfSurjective_of_kernelMulStandard_le
C hC hForm psi hpsi hker_mul
intro x
refine Submodule.Quotient.induction_on
(p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi) x ?_
intro y
let L :=
zcDiffToKerAugQuotOfSurj_of_kernelMulStandard_le
C hC hForm psi hpsi hker_mul
let P : zcCompletedGroupAlgebraStandardAugmentationIdeal C G → Prop := fun y =>
∃ m : ZCCompletedDifferentialModule C psi.toMonoidHom,
L m = Submodule.Quotient.mk
(p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi) y
have hy : P y := by
have hyspan : (y : ZCCompletedGroupAlgebra C G) ∈
Submodule.span (ZCCompletedGroupAlgebra C G)
(Set.range fun h : G => zcGroupLike C G h - 1) := by
change (y : ZCCompletedGroupAlgebra C G) ∈
zcCompletedGroupAlgebraStandardAugmentationIdeal C G
exact y.2
refine Submodule.span_induction
(p := fun z hz =>
P
⟨z, by
simpa [zcCompletedGroupAlgebraStandardAugmentationIdeal_eq_span] using hz⟩)
?hgen ?hzero ?hadd ?hsmul hyspan
· rintro _ ⟨g, rfl⟩
refine ⟨zcUniversalDifferential C psi.toMonoidHom g, ?_⟩
rw [zcDiffToKerAugQuotOfSurj_of_kernelMulStandard_le_universal]
rfl
· refine ⟨0, ?_⟩
change (0 : KernelAugmentationIdealQuotient C psi) =
Submodule.Quotient.mk
(p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi)
(0 : zcCompletedGroupAlgebraStandardAugmentationIdeal C G)
rw [Submodule.Quotient.mk_zero]
· intro a b ha hb hpa hpb
rcases hpa with ⟨ma, hma⟩
rcases hpb with ⟨mb, hmb⟩
refine ⟨ma + mb, ?_⟩
rw [map_add, hma, hmb, ← Submodule.Quotient.mk_add]
apply congrArg (fun t : zcCompletedGroupAlgebraStandardAugmentationIdeal C G =>
Submodule.Quotient.mk
(p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi) t)
exact Subtype.ext rfl
· intro a b hb hpb
rcases hpb with ⟨m, hm⟩
refine ⟨zcCompletedGroupAlgebraMap C hC psi a • m, ?_⟩
rw [map_smul, hm]
calc
zcCompletedGroupAlgebraMap C hC psi a •
Submodule.Quotient.mk
(p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi)
⟨b, by
simpa [zcCompletedGroupAlgebraStandardAugmentationIdeal_eq_span] using hb⟩ =
a •
Submodule.Quotient.mk
(p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi)
⟨b, by
simpa [zcCompletedGroupAlgebraStandardAugmentationIdeal_eq_span] using hb⟩ := by
exact
zcCompletedGAKerAugQuotTargetCompletedModuleOfSurjective_map_smul
C hC hForm psi hpsi hker_mul a
(Submodule.Quotient.mk
(p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi)
⟨b, by
simpa [zcCompletedGroupAlgebraStandardAugmentationIdeal_eq_span] using hb⟩)
_ = Submodule.Quotient.mk
(p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi)
⟨a • b, by
rw [zcCompletedGroupAlgebraStandardAugmentationIdeal_eq_span]
exact Submodule.smul_mem
(Submodule.span (ZCCompletedGroupAlgebra C G)
(Set.range fun h : G => zcGroupLike C G h - 1)) a hb⟩ := by
rw [← Submodule.Quotient.mk_smul]
apply congrArg (fun t : zcCompletedGroupAlgebraStandardAugmentationIdeal C G =>
Submodule.Quotient.mk
(p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi) t)
exact Subtype.ext rfl
exact hyProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□def zcCompletedGAKerAugQuotToClosedQuotientTargetLinear_of_kernelMulStandard_le
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(hker_mul :
∀ k : ZCCompletedGroupAlgebra C G,
k ∈ RingHom.ker (zcCompletedGroupAlgebraMap C hC psi) →
∀ y : zcCompletedGroupAlgebraStandardAugmentationIdeal C G,
(⟨k * (y : ZCCompletedGroupAlgebra C G),
(zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left k y.2⟩ :
zcCompletedGroupAlgebraStandardAugmentationIdeal C G) ∈
zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi) :
letI : Module (ZCCompletedGroupAlgebra C H) (KernelAugmentationIdealQuotient C psi) :=
zcCompletedGAKerAugQuotTargetCompletedModuleOfSurjective_of_kernelMulStandard_le
C hC hForm psi hpsi hker_mul
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
KernelAugmentationIdealQuotient C psi →ₗ[ZCCompletedGroupAlgebra C H]
KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen := by
letI : Module (ZCCompletedGroupAlgebra C H) (KernelAugmentationIdealQuotient C psi) :=
zcCompletedGAKerAugQuotTargetCompletedModuleOfSurjective_of_kernelMulStandard_le
C hC hForm psi hpsi hker_mul
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
let Q :=
zcCompletedGroupAlgebraKernelAugmentationQuotientToClosedQuotient
C hC hForm psi hpsi hfopen
refine
{ toFun := Q
map_add' := by
intro x y
exact map_add Q x y
map_smul' := by
intro a x
change Q
(zcCompletedGroupAlgebraTargetLiftOfSurjective C hC hForm psi hpsi a • x) =
a • Q x
rw [map_smul]
symm
calc
a • Q x =
zcCompletedGroupAlgebraMap C hC psi
(zcCompletedGroupAlgebraTargetLiftOfSurjective C hC hForm psi hpsi a) •
Q x := by
rw [zcCompletedGroupAlgebraMap_targetLiftOfSurjective]
_ =
zcCompletedGroupAlgebraTargetLiftOfSurjective C hC hForm psi hpsi a • Q x := by
exact
kerAugClosedQuotTargetCompletedModuleOfSurj_map_smul
C hC hForm psi hpsi hfopen
(zcCompletedGroupAlgebraTargetLiftOfSurjective C hC hForm psi hpsi a)
(Q x) }
@[simp 900]Under the explicit kernel-product hypothesis, the natural map from the algebraic source augmentation quotient to the closed quotient is \(\mathbb{Z}_C\llbracket H\rrbracket\)-linear.
theorem zcCompletedGAKerAugQuotToClosedQuotientTargetLinear_of_kernelMulStandard_le_mk
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(hker_mul :
∀ k : ZCCompletedGroupAlgebra C G,
k ∈ RingHom.ker (zcCompletedGroupAlgebraMap C hC psi) →
∀ y : zcCompletedGroupAlgebraStandardAugmentationIdeal C G,
(⟨k * (y : ZCCompletedGroupAlgebra C G),
(zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left k y.2⟩ :
zcCompletedGroupAlgebraStandardAugmentationIdeal C G) ∈
zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi)
(x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G) :
zcCompletedGAKerAugQuotToClosedQuotientTargetLinear_of_kernelMulStandard_le
C hC hForm psi hpsi hfopen hker_mul (Submodule.Quotient.mk x) =
(Submodule.Quotient.mk x :
KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)The natural map from the algebraic source augmentation quotient to the closed quotient sends each algebraic quotient class to its closed quotient class.
Show proof
by
letI : Module (ZCCompletedGroupAlgebra C H) (KernelAugmentationIdealQuotient C psi) :=
zcCompletedGAKerAugQuotTargetCompletedModuleOfSurjective_of_kernelMulStandard_le
C hC hForm psi hpsi hker_mul
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
rw [zcCompletedGAKerAugQuotToClosedQuotientTargetLinear_of_kernelMulStandard_le]
exact
zcCompletedGroupAlgebraKernelAugmentationQuotientToClosedQuotient_mk
C hC hForm psi hpsi hfopen xProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem zcDiffToKerAugClosedQuotOfSurj_eq_toClosed_comp_quotient_of_kernelMulStandard_le
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(hker_mul :
∀ k : ZCCompletedGroupAlgebra C G,
k ∈ RingHom.ker (zcCompletedGroupAlgebraMap C hC psi) →
∀ y : zcCompletedGroupAlgebraStandardAugmentationIdeal C G,
(⟨k * (y : ZCCompletedGroupAlgebra C G),
(zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left k y.2⟩ :
zcCompletedGroupAlgebraStandardAugmentationIdeal C G) ∈
zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi) :
(zcCompletedGAKerAugQuotToClosedQuotientTargetLinear_of_kernelMulStandard_le
C hC hForm psi hpsi hfopen hker_mul).comp
(zcDiffToKerAugQuotOfSurj_of_kernelMulStandard_le
C hC hForm psi hpsi hker_mul) =
zcCompletedDifferentialModuleToKernelAugmentationClosedQuotientOfSurjective
C hC hForm psi hpsi hfopenThe algebraic quotient map followed by the natural closed-quotient map is the closed quotient map already constructed directly from \(A_{\psi}(C)\).
Show proof
by
letI : Module (ZCCompletedGroupAlgebra C H) (KernelAugmentationIdealQuotient C psi) :=
zcCompletedGAKerAugQuotTargetCompletedModuleOfSurjective_of_kernelMulStandard_le
C hC hForm psi hpsi hker_mul
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
apply zcCompletedDifferentialModuleHom_ext C psi.toMonoidHom
intro g
calc
((zcCompletedGAKerAugQuotToClosedQuotientTargetLinear_of_kernelMulStandard_le
C hC hForm psi hpsi hfopen hker_mul).comp
(zcDiffToKerAugQuotOfSurj_of_kernelMulStandard_le
C hC hForm psi hpsi hker_mul))
(zcUniversalDifferential C psi.toMonoidHom g)
=
zcCompletedGAKerAugQuotToClosedQuotientTargetLinear_of_kernelMulStandard_le
C hC hForm psi hpsi hfopen hker_mul
(zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationQuotient C psi g) := by
rw [LinearMap.comp_apply,
zcDiffToKerAugQuotOfSurj_of_kernelMulStandard_le_universal]
_ =
zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
C hC hForm psi hpsi hfopen g := by
rw [zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationQuotient]
exact
zcCompletedGAKerAugQuotToClosedQuotientTargetLinear_of_kernelMulStandard_le_mk
C hC hForm psi hpsi hfopen hker_mul
⟨zcCompletedGroupAlgebraBoundary C (MonoidHom.id G) g,
zcCompletedGroupAlgebraBoundary_mem_standardAugmentationIdeal
C G (MonoidHom.id G) g⟩
_ =
zcCompletedDifferentialModuleToKernelAugmentationClosedQuotientOfSurjective
C hC hForm psi hpsi hfopen
(zcUniversalDifferential C psi.toMonoidHom g) := by
rw [zcDiffToKerAugClosedQuotOfSurj_universal]Proof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem sourceBoundaryToKerAug_isTargetCrossedDiff_of_surj_of_closed_kernelMulStandard
[Fact (ProCGroups.FiniteGroupClass.FiniteOnly C)]
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi)
(hfopen : IsOpenMap psi)
(hclosed :
IsClosed
((zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi :
Set (zcCompletedGroupAlgebraStandardAugmentationIdeal C G)))) :
letI : Module (ZCCompletedGroupAlgebra C H) (KernelAugmentationIdealQuotient C psi)If the product \(I(\ker \psi)I(G)\) is closed, the source boundary is a crossed differential for the descended completed target scalars.
Show proof
zcCompletedGAKerAugQuotTargetCompletedModuleOfSurjective_of_closed_kernelMulStandard
C hC hForm psi hpsi hfopen hclosed
IsCrossedDifferential
(zcCompletedGroupAlgebraScalar C psi.toMonoidHom)
(zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationQuotient C psi) := by
let hker_mul :=
zcCompletedGAKernelAugmentationIdealMulStandard_kernelMulStandard_le_of_isClosed
C hC hForm psi hpsi hfopen hclosed
letI : Module (ZCCompletedGroupAlgebra C H) (KernelAugmentationIdealQuotient C psi) :=
zcCompletedGAKerAugQuotTargetCompletedModuleOfSurjective_of_kernelMulStandard_le
C hC hForm psi hpsi hker_mul
exact
sourceBoundaryToKerAug_isTargetCrossedDiff_of_surj_of_kernelMulStandard_le
C hC hForm psi hpsi hker_mulProof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem zcCompletedDifferentialModule_sourceKernelGroupLikeSubOne_smul_eq_zero
(hC : ProCGroups.FiniteGroupClass.Hereditary C) (psi : ContinuousMonoidHom G H)
(n : ProfiniteKernelSubgroup psi) (x : ZCCompletedDifferentialModule C psi.toMonoidHom) :
letI : Module (ZCCompletedGroupAlgebra C G)
(ZCCompletedDifferentialModule C psi.toMonoidHom)Source kernel group-like differences act trivially on \(A_{\psi}(C)\) after restricting scalars along \(\mathbb{Z}_C\llbracket G\rrbracket \to \mathbb{Z}_C\llbracket H\rrbracket\).
Show proof
Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
(zcGroupLike C G n.1 - 1) • x = 0 := by
letI : Module (ZCCompletedGroupAlgebra C G)
(ZCCompletedDifferentialModule C psi.toMonoidHom) :=
Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
have hmap :
zcCompletedGroupAlgebraMap C hC psi (zcGroupLike C G n.1 - 1) = 0 := by
rw [map_sub, zcCompletedGroupAlgebraMap_groupLike, map_one]
rw [show psi (n : G) = 1 from n.2]
rw [map_one, sub_self]
change zcCompletedGroupAlgebraMap C hC psi (zcGroupLike C G n.1 - 1) • x = 0
rw [hmap, zero_smul]Proof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□theorem zcCompletedDifferentialModule_kernelAugmentationIdealMulStandard_smul_eq_zero
(hC : ProCGroups.FiniteGroupClass.Hereditary C) (psi : ContinuousMonoidHom G H)
(y : zcCompletedGroupAlgebraStandardAugmentationIdeal C G)
(hy : y ∈ zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi)
(x : ZCCompletedDifferentialModule C psi.toMonoidHom) :
letI : Module (ZCCompletedGroupAlgebra C G)
(ZCCompletedDifferentialModule C psi.toMonoidHom)The algebraic product \(I(\ker \psi)I(G)\) acts trivially on \(A_{\psi}(C)\) after restricting scalars along \(\mathbb{Z}_C\llbracket G\rrbracket \to \mathbb{Z}_C\llbracket H\rrbracket\).
Show proof
Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
(y : ZCCompletedGroupAlgebra C G) • x = 0 := by
letI : Module (ZCCompletedGroupAlgebra C G)
(ZCCompletedDifferentialModule C psi.toMonoidHom) :=
Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
change (y : ZCCompletedGroupAlgebra C G) • x = 0
refine Submodule.span_induction
(p := fun y : zcCompletedGroupAlgebraStandardAugmentationIdeal C G =>
fun _ => (y : ZCCompletedGroupAlgebra C G) • x = 0) ?_ ?_ ?_ ?_ hy
· rintro _ ⟨⟨n, s⟩, rfl⟩
change ((zcGroupLike C G n.1 - 1) * (s : ZCCompletedGroupAlgebra C G)) • x = 0
rw [mul_smul]
exact zcCompletedDifferentialModule_sourceKernelGroupLikeSubOne_smul_eq_zero
C hC psi n ((s : ZCCompletedGroupAlgebra C G) • x)
· change (0 : ZCCompletedGroupAlgebra C G) • x = 0
rw [zero_smul]
· intro y z _ _ hy hz
change ((y : ZCCompletedGroupAlgebra C G) +
(z : ZCCompletedGroupAlgebra C G)) • x = 0
rw [add_smul, hy, hz, zero_add]
· intro a y _ hy
change (a * (y : ZCCompletedGroupAlgebra C G)) • x = 0
rw [mul_smul, hy, smul_zero]Proof. Use the finite-stage closed augmentation quotient. The denominator is the finite-stage closed hull of the product of the kernel augmentation ideal with the standard augmentation ideal, so membership, closure, quotient, and continuity statements are tested after the finite-stage projections. Boundary and lift maps descend because they kill this denominator, and scalar actions are continuous because they are coordinatewise finite-stage module operations compatible with the quotient maps.
□