ProCGroups.FoxDifferential.Completed.Continuous.Universal.AugmentationQuotient
The principal declarations in this module are:
zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardThe algebraic product \(I(\ker \psi)I(G)\) inside the algebraic standard source augmentation ideal. -zcCompletedGroupAlgebraStandardAugmentationIdealProjectionProjection of the algebraic standard augmentation ideal to a finite augmentation stage. -zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard_generator_memA generator \((n-1)s\) lies in the algebraic product \(I(\ker \psi)I(G)\). -zcCompletedGroupAlgebraStandardAugmentationIdealProjection_valThe value of the augmentation-ideal projection is the value of the corresponding finite-stage projection.
imports
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.
omit [IsTopologicalGroup H] in
theorem zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard_generator_mem
(psi : ContinuousMonoidHom G H) (n : ProfiniteKernelSubgroup psi)
(s : zcCompletedGroupAlgebraStandardAugmentationIdeal C G) :
(zcGroupLike C G n.1 - 1) • s ∈
zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psiA generator \((n-1)s\) lies in the algebraic product \(I(\ker \psi)I(G)\).
Show Lean proof
Submodule.subset_span (Set.mem_range_self (n, s))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 xAugProjection of the algebraic standard augmentation ideal to a finite augmentation stage.
@[simp]
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 Lean proof
rfltheorem continuous_zcCompletedGroupAlgebraStandardAugmentationIdealProjection
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(i : ZCCompletedGroupAlgebraIndex C G) :
Continuous (zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i)The standard augmentation-ideal projection is continuous.
Show Lean 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)
Theorem
FoxDifferential.zcCompletedGAStageAugmentationIdeal_identityBoundary_monoidAlgebraToIdentity
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 Lean 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]
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 Lean proof
by
apply Subtype.ext
simp only [zcCompletedGroupAlgebraStandardAugmentationIdealProjection_val, ZeroMemClass.coe_zero,
zcCompletedGroupAlgebraProjection_zero]@[simp]
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 Lean proof
by
apply Subtype.ext
simp only [zcCompletedGroupAlgebraStandardAugmentationIdealProjection_val, Submodule.coe_add,
zcCompletedGroupAlgebraProjection_add]@[simp]
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 Lean 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]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 xThe finite-stage projection of the standard augmentation ideal, as a semilinear map over the completed group-algebra projection.
@[simp]
theorem zcCompletedGroupAlgebraStandardAugmentationIdealProjectionLinear_apply
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(i : ZCCompletedGroupAlgebraIndex C G)
(x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G) :
zcCompletedGroupAlgebraStandardAugmentationIdealProjectionLinear C i x =
zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i xThe linear finite-stage projection of the standard completed augmentation ideal has the same value as its underlying projection.
Show Lean proof
rfl@[simp]
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 Lean proof
by
simpa using
map_sub (zcCompletedGroupAlgebraStandardAugmentationIdealProjectionLinear C i) x ydef zcCompletedGroupAlgebraStandardAugmentationIdealProjectionProduct
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C] :
zcCompletedGroupAlgebraStandardAugmentationIdeal C G →
∀ i : ZCCompletedGroupAlgebraIndex C G,
zcCompletedGroupAlgebraStageAugmentationIdeal C G i :=
fun x i => zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i xThe product of all finite-stage projections of the standard completed augmentation ideal.
@[simp]
theorem zcCompletedGroupAlgebraStandardAugmentationIdealProjectionProduct_apply
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G)
(i : ZCCompletedGroupAlgebraIndex C G) :
zcCompletedGroupAlgebraStandardAugmentationIdealProjectionProduct C x i =
zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i xEvaluating the product of standard augmentation-ideal projections at i returns the projection at i.
Show Lean proof
rfltheorem zcCompletedGroupAlgebraStandardAugmentationIdealProjectionProduct_injective
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C] :
Function.Injective
(zcCompletedGroupAlgebraStandardAugmentationIdealProjectionProduct C (G := G))Finite-stage projections separate points of the standard completed augmentation ideal.
Show Lean proof
by
intro x y hxy
apply Subtype.ext
apply Subtype.ext
funext i
exact congrArg Subtype.val (congrFun hxy i)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 Lean proof
zcCompletedGroupAlgebraStandardAugmentationIdealProjectionProduct_injective C
(by
funext i
exact h i)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 Lean 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 hprojdef zcCompletedGroupAlgebraOpenImageKernelAugmentationIdealMulStageStandard
(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)The finite-stage product \(I(\ker)\,I(G/U)\) attached to the open-image quotient at a source stage.
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 iA finite-stage product generator belongs to the finite-stage product submodule.
Show Lean proof
Submodule.subset_span (Set.mem_range_self (q, s))def zcCompletedGroupAlgebraOpenImageKernelClass
(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]An actual kernel element determines a class in the finite-stage open-image kernel.
@[simp]
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 Lean proof
rfl
Theorem
FoxDifferential.zcCompletedGroupAlgebraStandardAugmentationIdealProjection_kernel_generator_smul
@[simp 900]
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 Lean 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]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 Lean 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 Lean 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]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 Lean proof
by
intro x hx i
exact
zcCompletedGAKernelAugmentationIdealMulStandard_proj_mem_openImageStage
C hC hForm psi hpsi hfopen i hxtheorem isClosed_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) :
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 Lean 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)theorem closure_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) :
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 Lean 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)theorem zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed_le_closure
[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 Lean 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) :=
Ssys.topologicalSpace
letI : ∀ i : ZCCompletedGroupAlgebraIndex C G, CompactSpace (Ssys.X i) := fun i => by
dsimp [Ssys, zcCompletedGroupAlgebraSystem]
change @CompactSpace (ZCCompletedGroupAlgebraStage C G i) ⊥
letI : Fact (0 < i.1.modulus) := ⟨i.1.positive⟩
letI : Finite (ZCCompletedGroupAlgebraStage C G i) :=
finite_modNCompletedGroupAlgebraStageInClass
(n := i.1.modulus) (G := G) C i.2
exact Finite.compactSpace
letI : ∀ i : ZCCompletedGroupAlgebraIndex C G, T2Space (Ssys.X i) := fun i => by
dsimp [Ssys, zcCompletedGroupAlgebraSystem]
change @T2Space (ZCCompletedGroupAlgebraStage C G i) ⊥
exact @DiscreteTopology.toT2Space _ ⊥ ⟨rfl⟩
have hdir : Directed (· ≤ ·) (id : ZCCompletedGroupAlgebraIndex C G →
ZCCompletedGroupAlgebraIndex C G) :=
directed_zcCompletedGroupAlgebraIndex_of_formation (C := C) (H := G) hForm
have hxLim :
(show Ssys.inverseLimit from (x : R)) ∈
closure (show Set Ssys.inverseLimit from Yamb) := by
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
convert hxLim using 1 <;> rfl
have hclosure :
closure Ystd =
(Subtype.val : zcCompletedGroupAlgebraStandardAugmentationIdeal C G → R) ⁻¹'
closure Yamb := by
exact Topology.IsEmbedding.subtypeVal.closure_eq_preimage_closure_image Ystd
change x ∈ closure Ystd
rw [hclosure]
exact hxAmbtheorem closure_zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard_eq_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) :
closure
((zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi :
Set (zcCompletedGroupAlgebraStandardAugmentationIdeal C G))) =
(zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen :
Set (zcCompletedGroupAlgebraStandardAugmentationIdeal C G))The closure of \(I(\ker \psi)I(G)\) is exactly the finite-stage kernel-product condition.
Show Lean 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)
Theorem
FoxDifferential.isClosed_zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard_iff_eq_closed
theorem isClosed_zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard_iff_eq_closed
{C : ProCGroups.FiniteGroupClass.{u}}
[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 Lean 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 hfopen
Abbreviation
FoxDifferential.KernelAugmentationIdealQuotient
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
[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
[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 Lean proof
by
rw [Topology.isQuotientMap_iff]
constructor
· exact ⟨rfl⟩
· exact
Submodule.Quotient.mk_surjective
(p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen)theorem continuous_kernelAugmentationIdealClosedQuotient_iff_comp_mkQ
{C : ProCGroups.FiniteGroupClass.{u}}
[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 Lean proof
by
simpa [Function.comp_def] using
(isQuotientMap_kernelAugmentationIdealClosedQuotient_mkQ
C hC hForm psi hpsi hfopen).continuous_iff (g := f)theorem t1Space_kernelAugmentationIdealClosedQuotient
[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 Lean 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_instancetheorem isClosed_zero_kernelAugmentationIdealClosedQuotient
[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 Lean proof
by
letI : T1Space
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
t1Space_kernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen
exact isClosed_singletonabbrev 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
[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)The finite-stage coordinate of the closed source augmentation quotient.
@[simp 900]
theorem kernelAugmentationIdealClosedQuotientStageProjection_mk
[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 Lean proof
by
exact
Submodule.mapQ_apply
(zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen)
(zcCompletedGroupAlgebraOpenImageKernelAugmentationIdealMulStageStandard
C hC hForm psi hpsi hfopen i)
(zcCompletedGroupAlgebraStandardAugmentationIdealProjectionLinear C i)
xtheorem continuous_kernelAugmentationIdealClosedQuotientStageProjection
[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 Lean proof
by
rw [continuous_kernelAugmentationIdealClosedQuotient_iff_comp_mkQ
(C := C) (G := G) (H := H) (hC := hC) (hForm := hForm)
(psi := psi) (hpsi := hpsi) (hfopen := 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'
have hcomp := hq.comp hproj
have hfun :
((fun y : zcCompletedGroupAlgebraStageAugmentationIdeal C G i =>
((zcCompletedGroupAlgebraOpenImageKernelAugmentationIdealMulStageStandard
C hC hForm psi hpsi hfopen i).mkQ y :
KernelAugmentationIdealClosedStageQuotient C hC hForm psi hpsi hfopen i)) ∘
zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i) =
(fun x : zcCompletedGroupAlgebraStandardAugmentationIdeal C G =>
kernelAugmentationIdealClosedQuotientStageProjection
C hC hForm psi hpsi hfopen i
((zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen).mkQ x)) := by
funext x
exact
(kernelAugmentationIdealClosedQuotientStageProjection_mk
C hC hForm psi hpsi hfopen i x).symm
exact hfun ▸ hcompdef kernelAugmentationIdealClosedQuotientStageProjectionProduct
[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 xThe product of all finite-stage coordinates of the closed source augmentation quotient.
@[simp]
theorem kernelAugmentationIdealClosedQuotientStageProjectionProduct_apply
[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 xEvaluating the product projection of the closed kernel-augmentation quotient at i returns its stage projection at i.
Show Lean proof
rfltheorem continuous_kernelAugmentationIdealClosedQuotientStageProjectionProduct
[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 Lean proof
by
exact continuous_pi fun i =>
continuous_kernelAugmentationIdealClosedQuotientStageProjection
C hC hForm psi hpsi hfopen itheorem kernelAugmentationIdealClosedQuotientStageProjectionProduct_injective
[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 Lean 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
change
kernelAugmentationIdealClosedQuotientStageProjection
C hC hForm psi hpsi hfopen i (S.mkQ x) =
kernelAugmentationIdealClosedQuotientStageProjection
C hC hForm psi hpsi hfopen i (S.mkQ y) at hi
rw [kernelAugmentationIdealClosedQuotientStageProjection_mk,
kernelAugmentationIdealClosedQuotientStageProjection_mk] at hi
exact hi
simpa [S, T] using hstagetheorem kernelAugmentationIdealClosedQuotientStageProjection_ext
[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 Lean proof
by
exact
kernelAugmentationIdealClosedQuotientStageProjectionProduct_injective
C hC hForm psi hpsi hfopen
(by
funext i
exact h i)theorem kernelAugmentationIdealClosedQuotient_topology_eq_induced_stageProjProduct
[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 Lean 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 hUquot :
@IsOpen Q (QuotientModule.Quotient.topologicalSpace Sclosed) U := hU
have hpreOpen :
@IsOpen
(zcCompletedGroupAlgebraStandardAugmentationIdeal C G)
inferInstance (q ⁻¹' U) := by
change
@IsOpen
(zcCompletedGroupAlgebraStandardAugmentationIdeal C G)
inferInstance (q ⁻¹' U) at hUquot
exact 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) :=
Ssys.topologicalSpace
letI : ∀ i : ZCCompletedGroupAlgebraIndex C G, CompactSpace (Ssys.X i) := fun i => by
dsimp [Ssys, zcCompletedGroupAlgebraSystem]
change @CompactSpace (ZCCompletedGroupAlgebraStage C G i) ⊥
letI : Fact (0 < i.1.modulus) := ⟨i.1.positive⟩
letI : Finite (ZCCompletedGroupAlgebraStage C G i) :=
finite_modNCompletedGroupAlgebraStageInClass
(n := i.1.modulus) (G := G) C i.2
exact Finite.compactSpace
letI : ∀ i : ZCCompletedGroupAlgebraIndex C G, T2Space (Ssys.X i) := fun i => by
dsimp [Ssys, zcCompletedGroupAlgebraSystem]
change @T2Space (ZCCompletedGroupAlgebraStage C G i) ⊥
exact @DiscreteTopology.toT2Space _ ⊥ ⟨rfl⟩
have hdir : Directed (· ≤ ·) (id : ZCCompletedGroupAlgebraIndex C G →
ZCCompletedGroupAlgebraIndex C G) :=
directed_zcCompletedGroupAlgebraIndex_of_formation (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
change
kernelAugmentationIdealClosedQuotientStageProjection
C hC hForm psi hpsi hfopen i (Sclosed.mkQ y) =
kernelAugmentationIdealClosedQuotientStageProjection
C hC hForm psi hpsi hfopen i (Sclosed.mkQ x) at hy'
rw [kernelAugmentationIdealClosedQuotientStageProjection_mk,
kernelAugmentationIdealClosedQuotientStageProjection_mk] at hy'
exact 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
have hi := (continuous_apply i).comp hprod
change Continuous
(kernelAugmentationIdealClosedQuotientStageProjection
C hC hForm psi hpsi hfopen i) at hi
exact hi
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)def zcCompletedGroupAlgebraOpenImageStageRingHom
(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)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.
@[simp]
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 Lean proof
by
simp only [zcCompletedGroupAlgebraOpenImageStageRingHom, MonoidAlgebra.of_apply]
exact MonoidAlgebra.mapDomain_singletheorem 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 Lean proof
by
intro y
change
MonoidAlgebra (ModNCompletedCoeff i.1.modulus)
(CompletedGroupAlgebraQuotientInClass H C
(zcCompletedGroupAlgebraOpenImageIndexInClass C hForm psi hpsi hfopen i)) at y
rcases
Finsupp.mapDomain_surjective (M := ModNCompletedCoeff i.1.modulus)
(zcCompletedGroupAlgebraOpenImageQuotientMap_surjective
C hC hForm psi hpsi hfopen i) y.coeff with
⟨x, hx⟩
refine ⟨MonoidAlgebra.ofCoeff x, ?_⟩
change MonoidAlgebra.mapDomain
(zcCompletedGroupAlgebraOpenImageQuotientMap
C hC hForm psi hpsi hfopen i) (MonoidAlgebra.ofCoeff x) = y
apply MonoidAlgebra.coeff_injective
simpa [MonoidAlgebra.mapDomain] using hxtheorem 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 Lean 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
change MonoidAlgebra.mapDomain f k = 0 at hk
exact 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 hb
Theorem
FoxDifferential.kernelAugmentationIdealClosedStageQuotient_openImageStageRingHom_ker_smul_eq_zero
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 Lean 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 s@[implicit_reducible]
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) :=
kernelAugmentationIdealClosedStageQuotientTargetStageModule
C hC hForm psi hpsi hfopen i
zcCompletedGroupAlgebraOpenImageStageRingHom C hC hForm psi hpsi hfopen i a • x =
a • xThe stage quotient target module map is compatible with scalar multiplication.
Show Lean proof
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 hzero@[implicit_reducible]
def zcCompletedDifferentialModuleOpenImageIndex
(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.
@[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 hForm psi hpsi hfopen i)) :
zcCompletedDifferentialModuleStageScalar C psi.toMonoidHom
(zcCompletedDifferentialModuleOpenImageIndex C 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 Lean proof
by
refine QuotientGroup.induction_on q ?_
intro g
simp only [zcCompletedDifferentialModuleStageScalar_coe,
zcCompletedGroupAlgebraOpenImageStageRingHom_of]
have hq :=
zcCompletedGroupAlgebraOpenImageQuotientMap_mk
C hC hForm psi hpsi hfopen i g
change
zcCompletedGroupAlgebraOpenImageQuotientMap C hC hForm psi hpsi hfopen i
(QuotientGroup.mk g :
CompletedGroupAlgebraQuotientInClass G C i.2) =
(QuotientGroup.mk (psi g) :
CompletedGroupAlgebraQuotientInClass H C
(zcCompletedGroupAlgebraOpenImageTargetIndex
C hForm psi hpsi hfopen i).2) at hq
symm
convert
congrArg
(MonoidAlgebra.of (ModNCompletedCoeff i.1.modulus)
(CompletedGroupAlgebraQuotientInClass H C
(zcCompletedGroupAlgebraOpenImageTargetIndex
C hForm psi hpsi hfopen i).2))
hq using 1 <;> rfldef 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) :
let j := zcCompletedDifferentialModuleOpenImageIndex C 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
ScalarCrossedHom
(zcCompletedDifferentialModuleStageScalar C psi.toMonoidHom j)
(KernelAugmentationIdealClosedStageQuotient C hC hForm psi hpsi hfopen i) := by
let j := zcCompletedDifferentialModuleOpenImageIndex C 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
refine
{ toFun := fun q =>
Submodule.Quotient.mk
(p := zcCompletedGroupAlgebraOpenImageKernelAugmentationIdealMulStageStandard
C hC hForm psi hpsi hfopen i)
(zcCompletedGroupAlgebraStageAugmentationGeneratorSubtype C G i q)
map_mul' := ?_ }
intro q r
rw [scalarCrossedAction_apply]
rw [zcCompletedDifferentialModuleOpenImageIndex_stageScalar
(C := C) (hC := hC) (hForm := hForm) (psi := psi)
(hpsi := hpsi) (hfopen := hfopen) (i := i)]
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 := Submodule.Quotient.mk
(p := zcCompletedGroupAlgebraOpenImageKernelAugmentationIdealMulStageStandard
C hC hForm psi hpsi hfopen i)
(zcCompletedGroupAlgebraStageAugmentationGeneratorSubtype C G 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)
change CompletedGroupAlgebraQuotientInClass G C i.2 at q r
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]
abelThe finite source-boundary coordinate over the matching open-image target stage.
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 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 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)The finite-stage lift induced by the finite closed source-boundary coordinate.
@[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 hForm psi hpsi hfopen i))
(a : zcCompletedDifferentialModuleStageRing C psi.toMonoidHom
(zcCompletedDifferentialModuleOpenImageIndex C hForm psi hpsi hfopen i)) :
let j := zcCompletedDifferentialModuleOpenImageIndex C 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 qThe boundary lift lands in the finite-stage closed augmentation quotient by the Fox-differential kernel condition.
Show Lean proof
by
let j := zcCompletedDifferentialModuleOpenImageIndex C 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]def zcCompletedGroupAlgebraKernelAugmentationQuotientToClosedQuotient
[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)The canonical quotient map from the algebraic kernel-product quotient to its closed finite-stage quotient.
@[simp]
theorem zcCompletedGroupAlgebraKernelAugmentationQuotientToClosedQuotient_mk
[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 Lean proof
by
rw [zcCompletedGroupAlgebraKernelAugmentationQuotientToClosedQuotient,
Submodule.mapQ_apply]
rfltheorem zcCompletedGAKerAugQuotToClosedQuotient_inj_iff_eq_closed
{C : ProCGroups.FiniteGroupClass.{u}}
[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))The canonical map from the algebraic quotient \(I(G) / I(\ker \psi)I(G)\) to the closed finite-stage quotient is injective exactly when the algebraic product already equals its finite-stage closed hull.
Show Lean 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 hmemSset
Theorem
FoxDifferential.isClosed_zcCompletedGAKernelAugmentationIdealMulStandard_iff_toClosedQuotient_inj
theorem isClosed_zcCompletedGAKernelAugmentationIdealMulStandard_iff_toClosedQuotient_inj
{C : ProCGroups.FiniteGroupClass.{u}}
[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 Lean proof
by
rw [isClosed_zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard_iff_eq_closed
(C := C) (psi := psi) hC hForm hpsi hfopen]
exact
(zcCompletedGAKerAugQuotToClosedQuotient_inj_iff_eq_closed
(C := C) (hC := hC) (hForm := hForm) (psi := psi)
(hpsi := hpsi) (hfopen := hfopen)).symmdef zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationQuotient
(psi : ContinuousMonoidHom G H) :
ScalarCrossedHom
(zcCompletedGroupAlgebraScalar C (MonoidHom.id G))
(KernelAugmentationIdealQuotient C psi) :=
(zcCompletedGroupAlgebraBoundaryToStandardAugmentationIdeal
C G (MonoidHom.id G)).mapLinear
(Submodule.mkQ
(zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi))The source Fox boundary, valued in the source augmentation quotient.
def zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
[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) :
ScalarCrossedHom
(zcCompletedGroupAlgebraScalar C (MonoidHom.id G))
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
(zcCompletedGroupAlgebraBoundaryToStandardAugmentationIdeal
C G (MonoidHom.id G)).mapLinear
(Submodule.mkQ
(zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen))The source Fox boundary, valued in the closed source augmentation quotient.
Theorem
FoxDifferential.zcCompletedGroupAlgebraKernelAugmentationQuotientToClosedQuotient_sourceBoundary
@[simp]
theorem zcCompletedGroupAlgebraKernelAugmentationQuotientToClosedQuotient_sourceBoundary
[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 Lean proof
by
simp only [zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationQuotient,
zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient,
ScalarCrossedHom.mapLinear_apply,
Submodule.mkQ_apply]
rw [
zcCompletedGroupAlgebraKernelAugmentationQuotientToClosedQuotient_mk]theorem continuous_zcCompletedGASourceBoundaryToKerAugClosedQuot
[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 Lean 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 hstdomit [IsTopologicalGroup H] in
@[simp]
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 Lean proof
by
apply (Submodule.Quotient.mk_eq_zero
(p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard C psi)).2
exact zcCompletedGroupAlgebraKernelAugmentationIdealMulStandard_generator_mem C psi n s
Theorem
FoxDifferential.zcCompletedGroupAlgebraKernelAugmentationQuotient_groupLike_smul_eq_of_map_eq
omit [IsTopologicalGroup H] in
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 Lean 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]
Definition
FoxDifferential.zcCompletedGroupAlgebraKernelAugmentationQuotientTargetGroupLikeEndOfSurjective
omit [IsTopologicalGroup H] in
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.
omit [IsTopologicalGroup H] in
@[simp]
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 Lean proof
rflomit [IsTopologicalGroup H] in
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 Lean 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 xomit [IsTopologicalGroup H] in
@[simp]
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 Lean 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 hsmulomit [IsTopologicalGroup H] in
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 Lean 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 hsmulomit [IsTopologicalGroup H] in
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 \(\psi\).
omit [IsTopologicalGroup H] in
@[simp]
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 Lean proof
rflomit [IsTopologicalGroup H] in
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.
omit [IsTopologicalGroup H] in
@[simp]
theorem zcCompletedGroupAlgebraKernelAugmentationQuotientTargetCoeffEnd_apply
(psi : ContinuousMonoidHom G H) (a : ZCCoeff C)
(x : KernelAugmentationIdealQuotient C psi) :
zcCompletedGroupAlgebraKernelAugmentationQuotientTargetCoeffEnd C psi a x =
zcCompletedGroupAlgebraCoeffMap C G a • xThe target coefficient endomorphism acts on the kernel-augmentation quotient by multiplication with the coefficient-map image of a.
Show Lean proof
rflomit [IsTopologicalGroup H] in
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.
omit [IsTopologicalGroup H] in
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 Lean 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 [mul_comm]omit [IsTopologicalGroup H] in
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.
omit [IsTopologicalGroup H] in
@[simp]
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 Lean proof
by
apply AddMonoidHom.ext
intro x
simp only [kerAugQuotTargetGAActionOfSurj, MonoidAlgebra.of_apply,
MonoidAlgebra.liftNCRingHom_single,
map_one, one_mul]
Abbreviation
FoxDifferential.zcCompletedGroupAlgebraKernelAugmentationQuotientTargetGroupAlgebraModuleOfSurjective
omit [IsTopologicalGroup H] in
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 descended algebraic Z_C[H]-module structure on the source augmentation quotient.
omit [IsTopologicalGroup H] in
@[simp]
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) :=
zcCompletedGroupAlgebraKernelAugmentationQuotientTargetGroupAlgebraModuleOfSurjective
C psi hpsi
a • x =
kerAugQuotTargetGAActionOfSurj
C psi hpsi a xThe target group-algebra module structure acts by the induced quotient action.
Show Lean proof
by
rflomit [IsTopologicalGroup H] in
@[simp]
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) :=
zcCompletedGroupAlgebraKernelAugmentationQuotientTargetGroupAlgebraModuleOfSurjective
C psi hpsi
MonoidAlgebra.of (ZCCoeff C) H h • x =
zcGroupLike C G (Function.surjInv hpsi h) • xA group-like basis element acts on the target quotient module by the induced quotient action.
Show Lean proof
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]
rflomit [IsTopologicalGroup H] in
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) :=
zcCompletedGroupAlgebraKernelAugmentationQuotientTargetGroupAlgebraModuleOfSurjective
C psi hpsi
MonoidAlgebra.of (ZCCoeff C) H h • x = zcGroupLike C G g • xThe algebraic target group-algebra action by \([h]\) agrees with source multiplication by any lift of \(h\).
Show Lean proof
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 xomit [IsTopologicalGroup H] in
def zcCompletedGASourceBoundaryToKerAugQuotTargetGACrossedHomOfSurjective
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi) :
letI : Module (MonoidAlgebra (ZCCoeff C) H) (KernelAugmentationIdealQuotient C psi) :=
zcCompletedGroupAlgebraKernelAugmentationQuotientTargetGroupAlgebraModuleOfSurjective
C psi hpsi
ScalarCrossedHom
((MonoidAlgebra.of (ZCCoeff C) H).comp psi.toMonoidHom)
(KernelAugmentationIdealQuotient C psi) := by
letI : Module (MonoidAlgebra (ZCCoeff C) H) (KernelAugmentationIdealQuotient C psi) :=
zcCompletedGroupAlgebraKernelAugmentationQuotientTargetGroupAlgebraModuleOfSurjective
C psi hpsi
exact
{ toFun := zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationQuotient C psi
map_mul' := by
intro g h
rw [(zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationQuotient
C psi).map_mul]
congr 1
exact
(kerAugQuotTargetGAModuleOfSurj_of_smul_eq_source_groupLike_smul
C psi hpsi (h := psi g) (g := g) rfl
(zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationQuotient
C psi h)).symm }The source boundary is a crossed differential for the descended algebraic target group-algebra coefficients.
omit [IsTopologicalGroup H] in
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
crossedHomModuleLift
(A := KernelAugmentationIdealQuotient C psi)
((MonoidAlgebra.of (ZCCoeff C) H).comp psi.toMonoidHom)
(zcCompletedGASourceBoundaryToKerAugQuotTargetGACrossedHomOfSurjective
C psi hpsi)The algebraic target-coefficient universal differential module maps to the source augmentation quotient by \(dg \mapsto\) \([g]-1\).
Theorem
FoxDifferential.zcAlgebraicDifferentialModuleToKernelAugmentationQuotientOfSurjective_universal
omit [IsTopologicalGroup H] in
@[simp]
theorem zcAlgebraicDifferentialModuleToKernelAugmentationQuotientOfSurjective_universal
(psi : ContinuousMonoidHom G H) (hpsi : Function.Surjective psi) (g : G) :
letI : Module (MonoidAlgebra (ZCCoeff C) H) (KernelAugmentationIdealQuotient C psi) :=
zcCompletedGroupAlgebraKernelAugmentationQuotientTargetGroupAlgebraModuleOfSurjective
C psi hpsi
zcAlgebraicDifferentialModuleToKernelAugmentationQuotientOfSurjective C psi hpsi
(universalCrossedDifferential
((MonoidAlgebra.of (ZCCoeff C) H).comp psi.toMonoidHom) g) =
zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationQuotient C psi gThe algebraic differential module maps universally to the kernel-augmentation quotient in the surjective case.
Show Lean proof
by
letI : Module (MonoidAlgebra (ZCCoeff C) H) (KernelAugmentationIdealQuotient C psi) :=
zcCompletedGroupAlgebraKernelAugmentationQuotientTargetGroupAlgebraModuleOfSurjective
C psi hpsi
exact
crossedHomModuleLift_universal
(A := KernelAugmentationIdealQuotient C psi)
((MonoidAlgebra.of (ZCCoeff C) H).comp psi.toMonoidHom)
(zcCompletedGASourceBoundaryToKerAugQuotTargetGACrossedHomOfSurjective
C psi hpsi) gomit [IsTopologicalGroup H] in
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 Lean 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, 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.coeff_add, 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]
Theorem
FoxDifferential.zcCompletedGAKernelAugmentationIdealMulStandard_mul_mem_closure_of_mem_ker_map
theorem zcCompletedGAKernelAugmentationIdealMulStandard_mul_mem_closure_of_mem_ker_map
[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 lies in the closure of the algebraic product; no closedness hypothesis is assumed.
Show Lean 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 hclosureImage
Theorem
FoxDifferential.zcCompletedGAKernelAugmentationIdealMulStandard_mul_mem_closed_of_mem_ker_map
theorem zcCompletedGAKernelAugmentationIdealMulStandard_mul_mem_closed_of_mem_ker_map
[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 Lean 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
[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 Lean proof
by
intro k hk y
exact
zcCompletedGAKernelAugmentationIdealMulStandard_mul_mem_closed_of_mem_ker_map
C hC hForm psi hpsi hfopen hk y
Theorem
FoxDifferential.zcCompletedGAKernelAugmentationIdealMulStandard_kernelMulStandard_le_of_isClosed
theorem zcCompletedGAKernelAugmentationIdealMulStandard_kernelMulStandard_le_of_isClosed
[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 Lean 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 hclosuretheorem kernelMulStandard_le_of_toClosedQuotient_inj
[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 Lean 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 hxzerotheorem 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 Lean 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 ydef zcCompletedGroupAlgebraTargetLiftOfSurjective
[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)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.
@[simp 900]
theorem zcCompletedGroupAlgebraMap_targetLiftOfSurjective
[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) = aFor a surjective presentation, the completed group-algebra map sends the chosen target lift of a back to a.
Show Lean proof
Function.surjInv_eq
(zcCompletedGroupAlgebraMap_surjective_of_surjective
(C := C) (hC := hC) hForm psi hpsi) a
Definition
FoxDifferential.zcCompletedGAKerAugQuotTargetCompletedModuleOfSurjective_of_kernelMulStandard_le
@[implicit_reducible]
def zcCompletedGAKerAugQuotTargetCompletedModuleOfSurjective_of_kernelMulStandard_le
[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 hzeroThe kernel-multiple containment hypothesis makes the source augmentation quotient a module over the target completed group algebra, using chosen lifts along the surjective presentation.
Definition
FoxDifferential.zcCompletedGAKerAugQuotTargetCompletedModuleOfSurjective_of_closed_kernelMulStandard
@[implicit_reducible]
def zcCompletedGAKerAugQuotTargetCompletedModuleOfSurjective_of_closed_kernelMulStandard
[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)Closedness of the kernel-multiple submodule supplies the containment needed to descend the target completed-group-algebra module structure to the source augmentation quotient.
Theorem
FoxDifferential.zcCompletedGroupAlgebraKernelAugmentationClosedQuotient_ker_map_smul_eq_zero
theorem zcCompletedGroupAlgebraKernelAugmentationClosedQuotient_ker_map_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)
(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 Lean 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 y@[implicit_reducible]
def kerAugClosedQuotTargetCompletedModuleOfSurj
[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
[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) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
zcCompletedGroupAlgebraMap C hC psi a • x = a • xThe closed quotient target module action is compatible with mapped scalars.
Show Lean proof
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 hzerotheorem continuous_zcCompletedGAKerAugClosedQuot_source_smul_const
[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 Lean 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'
change Continuous (fun a : ZCCompletedGroupAlgebra C G =>
(Submodule.Quotient.mk
(p := zcCompletedGroupAlgebraKernelAugmentationIdealMulStandardClosed
C hC hForm psi hpsi hfopen)
⟨a * (y : ZCCompletedGroupAlgebra C G),
(zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left a y.2⟩ :
KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen))
exact hq.comp hpretheorem continuous_kerAugClosedQuotTargetCompletedModuleOfSurj_smul_const
[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) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
Continuous (fun a : ZCCompletedGroupAlgebra C H => a • x)Descended target scalar multiplication on the closed source augmentation quotient is continuous in the target scalar, for a fixed quotient element.
Show Lean proof
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 hsource
Theorem
FoxDifferential.continuous_zcCompletedGroupAlgebraKernelAugmentationClosedQuotient_source_smul
theorem continuous_zcCompletedGroupAlgebraKernelAugmentationClosedQuotient_source_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) :
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 Lean 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'
change Continuous (fun p : ZCCompletedGroupAlgebra C G ×
zcCompletedGroupAlgebraStandardAugmentationIdeal C G =>
(Submodule.Quotient.mk (p := S)
⟨p.1 * (p.2 : ZCCompletedGroupAlgebra C G),
(zcCompletedGroupAlgebraStandardAugmentationIdeal C G).mul_mem_left p.1 p.2.2⟩ :
KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen))
exact hmk.comp hpretheorem continuous_kerAugClosedQuotTargetCompletedModuleOfSurj_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) :
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
Continuous (fun p : ZCCompletedGroupAlgebra C H ×
KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen =>
p.1 • p.2)Descended target scalar multiplication on the closed source augmentation quotient is jointly continuous.
Show Lean proof
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 hsource
Theorem
FoxDifferential.continuousSMul_zcCompletedGroupAlgebraKernelAugmentationClosedQuotient_source
theorem continuousSMul_zcCompletedGroupAlgebraKernelAugmentationClosedQuotient_source
[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 Lean proof
continuous_zcCompletedGroupAlgebraKernelAugmentationClosedQuotient_source_smul
C hC hForm psi hpsi hfopentheorem continuousSMul_kerAugClosedQuotTargetCompletedModuleOfSurj
[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
ContinuousSMul (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)The descended target module structure on the closed source augmentation quotient is topological.
Show Lean proof
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⟩theorem kerAugClosedQuotTargetCompletedModuleOfSurj_groupLike_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)
(g : G) (x : KernelAugmentationIdealClosedQuotient 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
zcGroupLike C H (psi g) • x = zcGroupLike C G g • xGroup-like elements act on the closed quotient target module by the induced target action.
Show Lean proof
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) x
Definition
FoxDifferential.zcCompletedGASourceBoundaryToKerAugClosedQuotTargetCompletedCrossedHomOfSurjective
def zcCompletedGASourceBoundaryToKerAugClosedQuotTargetCompletedCrossedHomOfSurjective
[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
ScalarCrossedHom
(zcCompletedGroupAlgebraScalar C psi.toMonoidHom)
(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
{ toFun :=
zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
C hC hForm psi hpsi hfopen
map_mul' := by
intro g h
rw [(zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
C hC hForm psi hpsi hfopen).map_mul]
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)).symm }The source boundary to the closed source augmentation quotient is a crossed differential for the descended completed target scalars.
theorem zcCompletedGroupAlgebraOpenImageStageRingHom_projection_targetLift
[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 Lean 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 hmaintheorem kerAugIdealClosedQuotStageProj_liftLinear_eq_boundaryLift_preStageMap
[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) :=
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 hForm psi hpsi hfopen i) x)The i-th closed augmentation quotient coordinate of the completed source-boundary lift factors through the corresponding open-image finite pre-stage.
Show Lean proof
by
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
let j := zcCompletedDifferentialModuleOpenImageIndex C 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
simp only [kernelAugmentationIdealClosedStageQuotientBoundary]
change
Submodule.Quotient.mk (p := T)
(zcCompletedGroupAlgebraStandardAugmentationIdealProjection C i s) =
Submodule.Quotient.mk (p := T)
(zcCompletedGroupAlgebraStageAugmentationGeneratorSubtype C G i
(zcCompletedDifferentialModuleStageSourceProj
C psi.toMonoidHom j g))
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
change
zcCompletedGroupAlgebraProjection C G i
(zcGroupLike C G g - 1) =
MonoidAlgebra.of (ModNCompletedCoeff i.1.modulus)
(CompletedGroupAlgebraQuotientInClass G C i.2)
(QuotientGroup.mk g) - 1
rw [zcCompletedGroupAlgebraProjection_sub,
zcCompletedGroupAlgebraProjection_groupLike,
zcCompletedGroupAlgebraProjection_one]
rw [hstage_boundary]
rfltheorem continuous_kernelAugmentationIdealClosedQuotientStageProjection_liftLinear
[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) :=
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))Each finite closed-augmentation coordinate of the pre-quotient source-boundary lift is continuous for the finite-stage pre-module topology.
Show Lean proof
by
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
let j := zcCompletedDifferentialModuleOpenImageIndex C 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 hpretheorem continuous_crossedDiffModuleLiftLinear_sourceBoundaryToKerAugClosedQuot
[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
@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))The pre-quotient source-boundary lift to the closed source augmentation quotient is continuous for the finite-stage pre-module topology.
Show Lean proof
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 idef zcSeparatedUniversalDifferentialSourceCompletedCrossedHom
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(psi : ContinuousMonoidHom G H) :
letI : Module (ZCCompletedGroupAlgebra C G)
(ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
ScalarCrossedHom
(zcCompletedGroupAlgebraScalar C (MonoidHom.id G))
(ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) := by
letI : Module (ZCCompletedGroupAlgebra C G)
(ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
exact
{ toFun := zcSeparatedUniversalDifferential C psi.toMonoidHom
map_mul' := by
intro g h
rw [(zcSeparatedUniversalDifferential C psi.toMonoidHom).map_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] }The separated universal differential is also a crossed differential for source completed group-algebra scalars after restricting scalars along \(\mathbb{Z}_C\llbracket G\rrbracket \to \mathbb{Z}_C\llbracket H\rrbracket\).
def zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule
[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
crossedHomModuleLift
(A := ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom)
(zcCompletedGroupAlgebraScalar C (MonoidHom.id G))
(zcSeparatedUniversalDifferentialSourceCompletedCrossedHom C hC psi)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 \(\mathbb{Z}_C\llbracket G\rrbracket\) \(\to\) \(\mathbb{Z}_C\llbracket H\rrbracket\).
@[simp]
theorem zcDiffModuleIdToZCSepDiffModule_universal
[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 Lean proof
by
letI : Module (ZCCompletedGroupAlgebra C G)
(ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
exact
crossedHomModuleLift_universal
(A := ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom)
(zcCompletedGroupAlgebraScalar C (MonoidHom.id G))
(zcSeparatedUniversalDifferentialSourceCompletedCrossedHom C hC psi) gtheorem zcCompletedDifferentialModuleIdentitySourceStageRingHom_transition_mapStage
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(psi : ContinuousMonoidHom G H)
(i : ZCCompletedDifferentialModuleIndex C psi.toMonoidHom) :
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
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 xThe finite source-identity coefficient map agrees with first projecting a completed source coefficient down to the source-identity stage and then applying the finite target map.
Show Lean proof
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
dsimp only [sourceIndex, idIndex, hle,
zcCompletedDifferentialModuleIdentitySourceIndex] at ⊢
rw [zcCompletedGroupAlgebraTransition_of]
simp only [MonoidAlgebra.of_apply]
simp only [zcCompletedGroupAlgebraMapStage,
zcCompletedDifferentialModuleIdentitySourceStageRingHom]
change MonoidAlgebra.mapDomain _ _ = MonoidAlgebra.mapDomain _ _
dsimp only [OpenNormalSubgroupInClass.map]
rw [QuotientGroup.map_mk]
dsimp only [CompletedGroupAlgebraQuotientInClass,
openNormalSubgroupInClassSystem] at ⊢
rw [MonoidAlgebra.mapDomain_single, MonoidAlgebra.mapDomain_single]
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
calc
zcCompletedGroupAlgebraMapStage C hC psi i.target
(zcCompletedGroupAlgebraTransition C G hle (x + y)) =
zcCompletedGroupAlgebraMapStage C hC psi i.target
(zcCompletedGroupAlgebraTransition C G hle x +
zcCompletedGroupAlgebraTransition C G hle y) := by
rw [map_add]
_ =
zcCompletedGroupAlgebraMapStage C hC psi i.target
(zcCompletedGroupAlgebraTransition C G hle x) +
zcCompletedGroupAlgebraMapStage C hC psi i.target
(zcCompletedGroupAlgebraTransition C G hle y) := by
rw [map_add]
_ =
zcCompletedDifferentialModuleIdentitySourceStageRingHom
C psi.toMonoidHom i x +
zcCompletedDifferentialModuleIdentitySourceStageRingHom
C psi.toMonoidHom i y := by
rw [hx, hy]
_ =
zcCompletedDifferentialModuleIdentitySourceStageRingHom
C psi.toMonoidHom i (x + y) := by
exact
(map_add
(zcCompletedDifferentialModuleIdentitySourceStageRingHom
C psi.toMonoidHom i) x y).symm
· 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 [zcCompletedGroupAlgebraMapStage,
zcCompletedGroupAlgebraTransition, modNCompletedGroupAlgebraStageCoeffMapInClass,
modNCompletedGroupRingCoeffMap, AlgHom.toRingHom_eq_coe, map_intCast]
symm
calc
zcCompletedDifferentialModuleIdentitySourceStageRingHom
C psi.toMonoidHom i (↑t * x) =
zcCompletedDifferentialModuleIdentitySourceStageRingHom
C psi.toMonoidHom i (↑t) *
zcCompletedDifferentialModuleIdentitySourceStageRingHom
C psi.toMonoidHom i x := by
exact map_mul _ _ _
_ = ↑t *
zcCompletedDifferentialModuleIdentitySourceStageRingHom
C psi.toMonoidHom i x := by
rw [map_intCast]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 Lean 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]theorem zcDiffModuleIdToZCSepDiffModule_stageProj
[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 Lean 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 crossedDifferentialModuleHom_ext
(zcCompletedGroupAlgebraScalar 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 xtheorem 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 Lean 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,
crossedHomModuleLift_universal, coefficientFoxBoundaryCrossedHom_apply]
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,
crossedHomModuleLift_universal, coefficientFoxBoundaryCrossedHom_apply]
rw [hboundaryTarget, hboundarySource]
rw [map_sub, map_one,
zcCompletedDifferentialModuleIdentitySourceStageRingHom_stageScalar]
exact LinearMap.congr_fun hLR xomit [IsTopologicalGroup H] in
theorem zcCompletedDifferentialModuleIdentitySourceStageBoundary_stageProj
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(psi : ContinuousMonoidHom G H)
(i : ZCCompletedDifferentialModuleIndex C psi.toMonoidHom)
(x : ZCCompletedDifferentialModule C (MonoidHom.id G)) :
let j := 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)Applying the finite identity boundary after the source-identity finite projection recovers the finite projection of the standard augmentation-valued completed Fox tail.
Show Lean proof
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 hcomptheorem zcDiffModuleIdToZCSepDiffModule_eq_zero_of_zcToStandard_eq_zero
[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 Lean 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]omit [IsTopologicalGroup G] in
theorem continuous_zcToStdAugIdeal_naturalTopology
(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 Lean 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
have hfun :
(fun x : ZCCompletedDifferentialModule C psi.toMonoidHom =>
(zcToStdAugIdeal C H psi.toMonoidHom x : ZCCompletedGroupAlgebra C H)) =
zcToCompletedGroupAlgebra C psi.toMonoidHom := by
funext x
exact zcToStdAugIdeal_val C H psi.toMonoidHom x
rw [hfun]
exact continuous_zcToCompletedGroupAlgebra_naturalTopology C hC psi
exact Continuous.subtype_mk hval
(fun x => (zcToStdAugIdeal
C H psi.toMonoidHom x).2)theorem zcSeparatedCompletedDifferentialModule_source_kernel_groupLike_smul
(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) :=
Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
zcGroupLike C G n.1 • x = xKernel 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 Lean proof
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]
rw [map_one]
exact one_smul _ xtheorem zcSeparatedCompletedDifferentialModule_source_map_smul
(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) :=
Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
zcCompletedGroupAlgebraMap C hC psi a • x = a • xSource scalar restriction on the separated module is exactly target scalar multiplication after applying the completed group-algebra map.
Show Lean proof
by
letI : Module (ZCCompletedGroupAlgebra C G)
(ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
rfltheorem zcSeparatedCompletedDifferentialModule_source_kernel_sub_one_smul
[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) :=
Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
(zcGroupLike C G n.1 - 1) • x = 0Elements of the source kernel augmentation ideal act trivially on the separated module after restricting scalars along the source map.
Show Lean proof
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]
exact sub_self xtheorem zcDiffModuleIdToZCSepDiffModule_kernel_sub_one_smul
[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 Lean 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)noncomputable def
stdAugIdealToZCSepDiffOfBoundaryKernel
[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.toLinearMapA conditional source-standard-augmentation map to the separated module. The only remaining well-definedness input is that the identity Fox tail kernel is killed by the source-identity lift to the separated module.
Definition
FoxDifferential.stdAugIdealToZCSepDiff
noncomputable def
stdAugIdealToZCSepDiff
[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)The source-standard-augmentation map to the separated module.
@[simp 900]
theorem
stdAugIdealToZCSepDiffOfBoundaryKernel_comp_zcToStdAugIdeal
[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)
(stdAugIdealToZCSepDiffOfBoundaryKernel
C hC psi hker).comp
(zcToStdAugIdeal C G (MonoidHom.id G)) =
zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule C hC psiThe conditional map out of the standard augmentation ideal recovers the canonical map from the completed differential module after composing with its quotient map.
Show Lean 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 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]
theorem
stdAugIdealToZCSepDiff_comp_zcToStdAugIdeal
[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)
(stdAugIdealToZCSepDiff
C hC psi).comp
(zcToStdAugIdeal C G (MonoidHom.id G)) =
zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule C hC psiThe standard augmentation-ideal map recovers the canonical separated differential-module map after precomposition with the quotient map.
Show Lean proof
by
exact
stdAugIdealToZCSepDiffOfBoundaryKernel_comp_zcToStdAugIdeal
C hC psi
(zcDiffModuleIdToZCSepDiffModule_eq_zero_of_zcToStandard_eq_zero
C hC psi)theorem
stdAugIdealToZCSepDiff_stageBoundary_stageProj
[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))Finite-stage boundary formula for the source-standard map into the separated module.
Show Lean 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 hmaptheorem
stdAugIdealToZCSepDiff_stageProj_eq_of_standardProj_eq
[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 finite-stage projection of the source-standard map depends only on the matching source-identity finite projection of the standard augmentation ideal.
Show Lean 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]theorem
continuous_stdAugIdealToZCSepDiff_stageProj
[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))Each finite-stage coordinate of the source-standard map into the separated module is continuous. The coordinate factors through the matching finite standard augmentation projection.
Show Lean 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)theorem
continuous_stdAugIdealToZCSepDiff_stageProjProduct
[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 product of all finite-stage coordinates of the source-standard map into the separated module is continuous.
Show Lean proof
by
exact continuous_pi fun i =>
continuous_stdAugIdealToZCSepDiff_stageProj
C hC psi itheorem
continuous_stdAugIdealToZCSepDiff
[ProCGroups.FiniteGroupClass.ContainsTrivialQuotients C]
(hC : ProCGroups.FiniteGroupClass.Hereditary C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(psi : ContinuousMonoidHom G H) :
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)The source-standard map to the separated completed differential module is continuous for the finite-stage quotient topology.
Show Lean proof
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]
theorem
stdAugIdealToZCSepDiffOfBoundaryKernel_boundary
[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 conditional augmentation-ideal map sends a group boundary to the separated universal differential of that group element.
Show Lean 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)
change
stdAugIdealToZCSepDiffOfBoundaryKernel C hC psi hker
(zcToStdAugIdeal C G (MonoidHom.id G)
(zcUniversalDifferential C (MonoidHom.id G) g)) =
zcCompletedDifferentialModuleIdToZCSeparatedCompletedDifferentialModule
C hC psi (zcUniversalDifferential C (MonoidHom.id G) g) at hcomp
have hboundary :
zcToStdAugIdeal C G (MonoidHom.id G)
(zcUniversalDifferential C (MonoidHom.id G) g) =
⟨zcCompletedGroupAlgebraBoundary C (MonoidHom.id G) g,
zcCompletedGroupAlgebraBoundary_mem_standardAugmentationIdeal
C G (MonoidHom.id G) g⟩ := by
exact
crossedHomModuleLift_universal
(A := zcCompletedGroupAlgebraStandardAugmentationIdeal C G)
(zcCompletedGroupAlgebraScalar C (MonoidHom.id G))
(zcCompletedGroupAlgebraBoundaryToStandardAugmentationIdeal
C G (MonoidHom.id G))
g
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
rw [← hboundary]
exact hcomp
_ = zcSeparatedUniversalDifferential C psi.toMonoidHom g := by
exact
zcDiffModuleIdToZCSepDiffModule_universal
C hC psi g@[simp]
theorem
stdAugIdealToZCSepDiff_boundary
[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 augmentation-ideal map sends a group boundary to the separated universal differential.
Show Lean proof
by
exact
stdAugIdealToZCSepDiffOfBoundaryKernel_boundary
C hC psi
(zcDiffModuleIdToZCSepDiffModule_eq_zero_of_zcToStandard_eq_zero
C hC psi) gtheorem
stdAugIdealToZCSepDiffOfBoundaryKernel_kernel_generator_smul
[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 conditional source-standard map kills each algebraic kernel-product generator.
Show Lean 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)
change
M (f ((zcGroupLike C G n.1 - 1) • y)) =
L ((zcGroupLike C G n.1 - 1) • y) at hcomp
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
exact hcomp
_ = 0 := by
exact
zcDiffModuleIdToZCSepDiffModule_kernel_sub_one_smul
C hC psi n ytheorem
stdAugIdealToZCSepDiffOfBoundaryKernel_kills_kernelMulStandard
[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 conditional source-standard map kills the algebraic product I(ker psi) I(G).
Show Lean 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
· exact 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]theorem
stdAugIdealToZCSepDiff_kills_kernelMulStandard
[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 source-standard map kills the algebraic product I(ker psi) I(G).
Show Lean proof
stdAugIdealToZCSepDiffOfBoundaryKernel_kills_kernelMulStandard
C hC psi
(zcDiffModuleIdToZCSepDiffModule_eq_zero_of_zcToStandard_eq_zero
C hC psi) hx
Theorem
FoxDifferential.stdAugIdealToZCSepDiffOfBoundaryKernel_kills_kernelMulStandardClosed_of_continuous
theorem
stdAugIdealToZCSepDiffOfBoundaryKernel_kills_kernelMulStandardClosed_of_continuous
[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 = 0If the conditional source-standard map is continuous for the separated quotient topology, then killing the algebraic denominator implies that it kills the finite-stage closed denominator.
Show Lean 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 hxcltheorem
stdAugIdealToZCSepDiff_kills_kernelMulStandardClosed_of_continuous
[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 = 0If the source-standard map is continuous for the separated quotient topology, then it kills the closed finite-stage denominator.
Show Lean 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' hxtheorem
stdAugIdealToZCSepDiff_kills_kernelMulStandardClosed
[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 source-standard map kills the closed finite-stage denominator.
Show Lean proof
stdAugIdealToZCSepDiff_kills_kernelMulStandardClosed_of_continuous
C hC hForm psi hpsi hfopen
(continuous_stdAugIdealToZCSepDiff
C hC hForm psi) hxnoncomputable def
kerAugIdealQuotToZCSepDiffOfBoundaryKernelOfClosedKill
[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)A conditional reverse map from the closed source augmentation quotient to the separated module. The remaining closed-denominator input is isolated as hclosed_kill.
@[simp 900]
theorem
kerAugIdealQuotToZCSepDiffOfBoundaryKernelOfClosedKill_boundary
[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 conditional map from the closed augmentation quotient sends the class of a boundary to the separated universal differential.
Show Lean 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 gnoncomputable def
kerAugIdealQuotToZCSepDiffLinearOfBoundaryKernelOfClosedKill
[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) }Target-linear version of the conditional reverse map from the closed source augmentation quotient to the separated module.
@[simp 900]
theorem
kerAugIdealQuotToZCSepDiffLinearOfBoundaryKernelOfClosedKill_boundary
[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 target-linear conditional quotient map sends each boundary class to the separated universal differential.
Show Lean proof
by
exact
kerAugIdealQuotToZCSepDiffOfBoundaryKernelOfClosedKill_boundary
C hC hForm psi hpsi hfopen hker hclosed_kill gnoncomputable def
kerAugIdealQuotToZCSepDiffLinearOfBoundaryKernelOfContStdMap
[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)Target-linear reverse map from the closed source augmentation quotient to the separated module, reducing the closed-denominator condition to continuity of the source-standard map.
@[simp 900]
theorem
kerAugIdealQuotToZCSepDiffLinearOfBoundaryKernelOfContStdMap_boundary
[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 gUnder continuity of the conditional standard map, the target-linear quotient map sends a boundary class to the separated universal differential.
Show Lean 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)
gnoncomputable def
kerAugIdealQuotToZCSepDiffOfClosedKill
[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)Reverse map from the closed source augmentation quotient to the separated module, assuming only that the closed denominator is killed by the unconditional source-standard map.
@[simp 900]
theorem
kerAugIdealQuotToZCSepDiffOfClosedKill_boundary
[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 gIf the closed denominator is killed, the resulting quotient map sends a boundary class to the separated universal differential.
Show Lean 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) gtheorem
continuous_kerAugIdealQuotToZCSepDiffOfClosedKill
[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) :=
zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom
Continuous
(kerAugIdealQuotToZCSepDiffOfClosedKill
C hC hForm psi hpsi hfopen hclosed_kill)The reverse map from the closed source augmentation quotient to the separated module is continuous once the source-standard map is continuous.
Show Lean proof
by
letI : TopologicalSpace (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom
rw [continuous_kernelAugmentationIdealClosedQuotient_iff_comp_mkQ
(C := C) (hC := hC) (hForm := hForm) (psi := psi)
(hpsi := hpsi) (hfopen := 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
rfl
rw [hcomp]
exact hcontnoncomputable def
kerAugIdealQuotToZCSepDiffLinearOfClosedKill
[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)Target-linear reverse map from the closed source augmentation quotient to the separated module, assuming only that the closed denominator is killed by the unconditional source-standard map.
@[simp 900]
theorem
kerAugIdealQuotToZCSepDiffLinearOfClosedKill_boundary
[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 target-linear map induced by annihilating the closed denominator has the expected value on boundary classes.
Show Lean proof
kerAugIdealQuotToZCSepDiffOfClosedKill_boundary
C hC hForm psi hpsi hfopen hclosed_kill gnoncomputable def
kerAugIdealQuotToZCSepDiffLinearOfContStdMap
[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)Target-linear reverse map from the closed source augmentation quotient to the separated module, reducing the closed-denominator condition to continuity of the unconditional source-standard map.
@[simp 900]
theorem
kerAugIdealQuotToZCSepDiffLinearOfContStdMap_boundary
[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 gContinuity of the standard map gives a target-linear quotient map with the expected value on boundary classes.
Show Lean proof
kerAugIdealQuotToZCSepDiffLinearOfClosedKill_boundary
C hC hForm psi hpsi hfopen
(fun _ hx =>
stdAugIdealToZCSepDiff_kills_kernelMulStandardClosed_of_continuous
C hC hForm psi hpsi hfopen hcont hx) gtheorem
continuous_kerAugIdealQuotToZCSepDiffLinearOfContStdMap
[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) :=
zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom
Continuous
(kerAugIdealQuotToZCSepDiffLinearOfContStdMap
C hC hForm psi hpsi hfopen hcont)The target-linear reverse map obtained from a continuous source-standard map is continuous.
Show Lean proof
by
letI : TopologicalSpace (ZCSeparatedCompletedDifferentialModule C psi.toMonoidHom) :=
zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom
change Continuous
(kerAugIdealQuotToZCSepDiffOfClosedKill
C hC hForm psi hpsi hfopen
(fun _ hx =>
stdAugIdealToZCSepDiff_kills_kernelMulStandardClosed_of_continuous
C hC hForm psi hpsi hfopen hcont hx))
exact
continuous_kerAugIdealQuotToZCSepDiffOfClosedKill
C hC hForm psi hpsi hfopen
(fun _ hx =>
stdAugIdealToZCSepDiff_kills_kernelMulStandardClosed_of_continuous
C hC hForm psi hpsi hfopen hcont hx)
hcontnoncomputable def
kerAugIdealQuotToZCSepDiffLinear
[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)Target-linear reverse map from the closed source augmentation quotient to the separated module.
@[simp 900]
theorem
kerAugIdealQuotToZCSepDiffLinear_boundary
[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 canonical target-linear reverse map sends each boundary class to the separated universal differential.
Show Lean proof
kerAugIdealQuotToZCSepDiffLinearOfContStdMap_boundary
C hC hForm psi hpsi hfopen
(continuous_stdAugIdealToZCSepDiff
C hC hForm psi) gtheorem
continuous_kerAugIdealQuotToZCSepDiffLinear
[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) :=
zcSeparatedCompletedDifferentialModuleNaturalTopology C psi.toMonoidHom
Continuous
(kerAugIdealQuotToZCSepDiffLinear
C hC hForm psi hpsi hfopen)The target-linear reverse map from the closed source augmentation quotient to the separated module is continuous.
Show Lean proof
continuous_kerAugIdealQuotToZCSepDiffLinearOfContStdMap
C hC hForm psi hpsi hfopen
(continuous_stdAugIdealToZCSepDiff
C hC hForm psi)
Definition
FoxDifferential.zcCompletedDifferentialModuleToKernelAugmentationClosedQuotientOfSurjective
def zcCompletedDifferentialModuleToKernelAugmentationClosedQuotientOfSurjective
[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
crossedHomModuleLift
(A := KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)
(zcCompletedGroupAlgebraScalar C psi.toMonoidHom)
(zcCompletedGASourceBoundaryToKerAugClosedQuotTargetCompletedCrossedHomOfSurjective
C hC hForm psi hpsi hfopen)The completed universal differential module maps to the closed source augmentation quotient by \(dg \mapsto\) \([g]-1\).
@[simp 900]
theorem zcDiffToKerAugClosedQuotOfSurj_universal
[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 Lean proof
by
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
exact
crossedHomModuleLift_universal
(A := KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen)
(zcCompletedGroupAlgebraScalar C psi.toMonoidHom)
(zcCompletedGASourceBoundaryToKerAugClosedQuotTargetCompletedCrossedHomOfSurjective
C hC hForm psi hpsi hfopen) g
Theorem
FoxDifferential.zcDiffToKerAugClosedQuotOfSurj_kills_finiteClosedSubmodule_of_continuous_lift
theorem zcDiffToKerAugClosedQuotOfSurj_kills_finiteClosedSubmodule_of_continuous_lift
[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) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
crossedDifferentialModuleLiftLinear
(R := ZCCompletedGroupAlgebra C H)
(zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
C hC hForm psi hpsi hfopen) x = 0If 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 Lean proof
by
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
letI : T1Space
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
t1Space_kernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen
exact
crossedDifferentialModuleLiftLinear_kills_finiteClosedSubmodule_of_continuous
C psi.toMonoidHom hdir
(zcCompletedGASourceBoundaryToKerAugClosedQuotTargetCompletedCrossedHomOfSurjective
C hC hForm psi hpsi hfopen)
hcont hxtheorem zcDiffToKerAugClosedQuotOfSurj_kills_finiteClosedSubmodule_of_continuous_lift_of_surj
[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) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
crossedDifferentialModuleLiftLinear
(R := ZCCompletedGroupAlgebra C H)
(zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationClosedQuotient
C hC hForm psi hpsi hfopen) x = 0Version of zcDiffToKerAugClosedQuotOfSurj_kills_finiteClosedSubmodule_of_continuous_lift with nonemptiness and directedness of finite stages supplied by the continuous source map.
Show Lean proof
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 hxdef zcSepDiffToKerAugClosedQuotOfSurjective
[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
letI : T1Space
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
t1Space_kernelAugmentationIdealClosedQuotient
C hC hForm psi hpsi hfopen
exact
zcSeparatedCompletedDifferentialModuleLiftOfContinuousPrelift
C psi.toMonoidHom hdir
(zcCompletedGASourceBoundaryToKerAugClosedQuotTargetCompletedCrossedHomOfSurjective
C hC hForm psi hpsi hfopen)
hcontUnder 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
[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)
hcontPublic version of the closed-augmentation descent map with finite-stage nonemptiness and directedness supplied by the continuous source map.
@[simp 900]
theorem zcSepDiffToKerAugClosedQuotOfSurjective_universal
[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 Lean proof
by
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
letI : T1Space
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
t1Space_kernelAugmentationIdealClosedQuotient
C hC hForm psi hpsi hfopen
exact
zcSeparatedCompletedDifferentialModuleLiftOfContinuousPrelift_universal
C psi.toMonoidHom hdir
(zcCompletedGASourceBoundaryToKerAugClosedQuotTargetCompletedCrossedHomOfSurjective
C hC hForm psi hpsi hfopen)
hcont g@[simp 900]
theorem zcSepDiffToKerAugClosedQuotOfSurjectiveOfContinuousLift_universal
[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 Lean 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 gtheorem zcSepDiffToKerAugClosedQuotOfSurjective_comp_toSep
[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
(zcSepDiffToKerAugClosedQuotOfSurjective
C hC hForm psi hpsi hfopen hdir hcont).comp
(zcCompletedDifferentialModuleToSeparated C psi.toMonoidHom) =
zcCompletedDifferentialModuleToKernelAugmentationClosedQuotientOfSurjective
C hC hForm psi hpsi hfopenThe separated closed-augmentation map is the factorization of the algebraic closed-augmentation map through \(A_{\psi}(C) \to A_{\psi}(C)_{\mathrm{sep}}\).
Show Lean proof
by
letI : Module (ZCCompletedGroupAlgebra C H)
(KernelAugmentationIdealClosedQuotient C hC hForm psi hpsi hfopen) :=
kerAugClosedQuotTargetCompletedModuleOfSurj
C hC hForm psi hpsi hfopen
apply crossedDifferentialModuleHom_ext
(zcCompletedGroupAlgebraScalar C psi.toMonoidHom)
intro g
change
zcSepDiffToKerAugClosedQuotOfSurjective
C hC hForm psi hpsi hfopen hdir hcont
(zcCompletedDifferentialModuleToSeparated C psi.toMonoidHom
(zcUniversalDifferential C psi.toMonoidHom g)) =
zcCompletedDifferentialModuleToKernelAugmentationClosedQuotientOfSurjective
C hC hForm psi hpsi hfopen
(zcUniversalDifferential C psi.toMonoidHom g)
rw [zcCompletedDifferentialModuleToSeparated_universal,
zcSepDiffToKerAugClosedQuotOfSurjective_universal,
zcDiffToKerAugClosedQuotOfSurj_universal]theorem zcSepDiffToKerAugClosedQuotOfSurjectiveOfContinuousLift_comp_toSep
[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
(zcSepDiffToKerAugClosedQuotOfSurjectiveOfContinuousLift
C hC hForm psi hpsi hfopen hcont).comp
(zcCompletedDifferentialModuleToSeparated C psi.toMonoidHom) =
zcCompletedDifferentialModuleToKernelAugmentationClosedQuotientOfSurjective
C hC hForm psi hpsi hfopenThe continuous-lift map to the closed kernel-augmentation quotient composes with separation as expected.
Show Lean proof
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)
hconttheorem continuous_zcSepDiffToKerAugClosedQuotOfSurjective
[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) :=
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)The descended forward map to the closed augmentation quotient is continuous once the pre-quotient source-boundary lift is continuous.
Show Lean proof
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,
zcSeparatedCompletedDifferentialModuleLiftOfContinuousPrelift,
Submodule.mkQ_apply, Submodule.liftQ_apply]
rfl
rw [hcomp]
exact hconttheorem continuous_zcSepDiffToKerAugClosedQuotOfSurjectiveOfContinuousLift
[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) :=
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)Continuity of the forward map when the finite-stage index data are supplied by the continuous source map.
Show Lean proof
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)
hconttheorem zcDiffToKerAugClosedQuotOfSurj_surj
[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 Lean 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 hytheorem zcSepDiffToKerAugClosedQuotOfSurjective_surj
[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 Lean 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)
change
zcSepDiffToKerAugClosedQuotOfSurjective
C hC hForm psi hpsi hfopen hdir hcont
(zcCompletedDifferentialModuleToSeparated C psi.toMonoidHom m) =
zcCompletedDifferentialModuleToKernelAugmentationClosedQuotientOfSurjective
C hC hForm psi hpsi hfopen m at hfactor
exact hfactor.trans hmtheorem zcSepDiffToKerAugClosedQuotOfSurjectiveOfContinuousLift_surj
[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 Lean 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)
hcontdef zcSeparatedCompletedDifferentialModuleToKernelAugmentationClosedQuotient
[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 hcontThe 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.
@[simp]
theorem zcSepDiffToKerAugClosedQuot_universal
[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 Lean 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 gtheorem zcSepDiffToKerAugClosedQuot_comp_toSep
[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
(zcSeparatedCompletedDifferentialModuleToKernelAugmentationClosedQuotient
C hC hForm psi hpsi hfopen).comp
(zcCompletedDifferentialModuleToSeparated C psi.toMonoidHom) =
zcCompletedDifferentialModuleToKernelAugmentationClosedQuotientOfSurjective
C hC hForm psi hpsi hfopenThe unconditional separated closed-augmentation map factors the algebraic map through \(A_{\psi}(C) \to A_{\psi}(C)_{\mathrm{sep}}\).
Show Lean proof
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 hconttheorem continuous_zcSepDiffToKerAugClosedQuot
[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) :=
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)The separated closed-augmentation map is continuous, with the pre-quotient lift continuity provided by the finite-stage factorization theorem.
Show Lean proof
by
let hcont :=
continuous_crossedDiffModuleLiftLinear_sourceBoundaryToKerAugClosedQuot
C hC hForm psi hpsi hfopen
simpa [zcSeparatedCompletedDifferentialModuleToKernelAugmentationClosedQuotient, hcont] using
continuous_zcSepDiffToKerAugClosedQuotOfSurjectiveOfContinuousLift
C hC hForm psi hpsi hfopen hconttheorem zcSepDiffToKerAugClosedQuot_surj
[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 Lean 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]
theorem kerAugIdealQuotToZCSepDiffLinear_mk
[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 Lean proof
by
rfltheorem kerAugIdealQuotToZCSepDiffLinear_comp_zcSepDiffToKerAugClosedQuot
[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
(kerAugIdealQuotToZCSepDiffLinear
C hC hForm psi hpsi hfopen).comp
(zcSeparatedCompletedDifferentialModuleToKernelAugmentationClosedQuotient
C hC hForm psi hpsi hfopen) =
LinearMap.idComposing the reverse closed-augmentation map with the forward separated map gives the identity.
Show Lean proof
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]
rw [LinearMap.id_apply]
change
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 g
exact
kerAugIdealQuotToZCSepDiffLinear_boundary
C hC hForm psi hpsi hfopen gtheorem zcSepDiffToKerAugClosedQuot_comp_kerAugIdealQuotToZCSepDiffLinear
[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
(zcSeparatedCompletedDifferentialModuleToKernelAugmentationClosedQuotient
C hC hForm psi hpsi hfopen).comp
(kerAugIdealQuotToZCSepDiffLinear
C hC hForm psi hpsi hfopen) =
LinearMap.idComposing the forward separated map with the reverse closed-augmentation map gives the identity.
Show Lean proof
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
change
(y' : ZCCompletedGroupAlgebra C G) =
zcGroupLike C G g - 1
exact 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)
change F (S 0) = 0
rw [map_zero, map_zero])
(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 rfldef zcSepDiffEquivKerAugClosedQuot_of_surj
[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)The separated completed differential module is the closed source augmentation quotient for a surjective open continuous homomorphism.
@[simp]
theorem zcSepDiffEquivKerAugClosedQuot_of_surj_apply
[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 Lean proof
rfl@[simp]
theorem zcSepDiffEquivKerAugClosedQuot_of_surj_symm_apply
[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 Lean proof
rfldef zcApsiEquivKerAugClosedQuot_of_surj
[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 hfopenThe 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.
@[simp]
theorem zcApsiEquivKerAugClosedQuot_of_surj_apply
[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 Lean proof
rfl@[simp]
theorem zcApsiEquivKerAugClosedQuot_of_surj_symm_apply
[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 Lean proof
rfltheorem zcCompletedGAKerAugQuotTargetCompletedModuleOfSurjective_map_smul
[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) :=
zcCompletedGAKerAugQuotTargetCompletedModuleOfSurjective_of_kernelMulStandard_le
C hC hForm psi hpsi hker_mul
zcCompletedGroupAlgebraMap C hC psi a • x = a • xFor a surjective target map, the completed target scalar action on the kernel-augmentation quotient is compatible with applying the induced algebra map.
Show Lean proof
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 hzerotheorem zcCompletedGAKerAugQuotTargetCompletedModuleOfSurjective_groupLike_smul
[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) :=
zcCompletedGAKerAugQuotTargetCompletedModuleOfSurjective_of_kernelMulStandard_le
C hC hForm psi hpsi hker_mul
zcGroupLike C H (psi g) • x = zcGroupLike C G g • xIn the descended target module structure, the group-like element of \(\psi(g)\) acts in the same way as the source group-like element of \(g\).
Show Lean proof
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) xdef sourceBoundaryToKerAugTargetCrossedHomOfSurjectiveOfKernelMulStandardLe
[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
ScalarCrossedHom
(zcCompletedGroupAlgebraScalar C psi.toMonoidHom)
(KernelAugmentationIdealQuotient C psi) := by
letI : Module (ZCCompletedGroupAlgebra C H) (KernelAugmentationIdealQuotient C psi) :=
zcCompletedGAKerAugQuotTargetCompletedModuleOfSurjective_of_kernelMulStandard_le
C hC hForm psi hpsi hker_mul
exact
{ toFun := zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationQuotient C psi
map_mul' := by
intro g h
rw [(zcCompletedGroupAlgebraSourceBoundaryToKernelAugmentationQuotient
C psi).map_mul]
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)).symm }Under the explicit kernel-product hypothesis, the source boundary is a crossed differential for the descended completed target scalars.
def zcDiffToKerAugQuotOfSurj_of_kernelMulStandard_le
[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
crossedHomModuleLift
(A := KernelAugmentationIdealQuotient C psi)
(zcCompletedGroupAlgebraScalar C psi.toMonoidHom)
(sourceBoundaryToKerAugTargetCrossedHomOfSurjectiveOfKernelMulStandardLe
C hC hForm psi hpsi hker_mul)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.
@[simp 900]
theorem zcDiffToKerAugQuotOfSurj_of_kernelMulStandard_le_universal
[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 Lean proof
by
letI : Module (ZCCompletedGroupAlgebra C H) (KernelAugmentationIdealQuotient C psi) :=
zcCompletedGAKerAugQuotTargetCompletedModuleOfSurjective_of_kernelMulStandard_le
C hC hForm psi hpsi hker_mul
exact
crossedHomModuleLift_universal
(A := KernelAugmentationIdealQuotient C psi)
(zcCompletedGroupAlgebraScalar C psi.toMonoidHom)
(sourceBoundaryToKerAugTargetCrossedHomOfSurjectiveOfKernelMulStandardLe
C hC hForm psi hpsi hker_mul) gtheorem zcDiffToKerAugQuotOfSurj_of_kernelMulStandard_le_surj
[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 Lean 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 hy
Definition
FoxDifferential.zcCompletedGAKerAugQuotToClosedQuotientTargetLinear_of_kernelMulStandard_le
def zcCompletedGAKerAugQuotToClosedQuotientTargetLinear_of_kernelMulStandard_le
[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) }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
FoxDifferential.zcCompletedGAKerAugQuotToClosedQuotientTargetLinear_of_kernelMulStandard_le_mk
@[simp 900]
theorem zcCompletedGAKerAugQuotToClosedQuotientTargetLinear_of_kernelMulStandard_le_mk
[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 Lean 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 x
Theorem
FoxDifferential.zcDiffToKerAugClosedQuotOfSurj_eq_toClosed_comp_quotient_of_kernelMulStandard_le
theorem zcDiffToKerAugClosedQuotOfSurj_eq_toClosed_comp_quotient_of_kernelMulStandard_le
[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 Lean 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 crossedDifferentialModuleHom_ext
(zcCompletedGroupAlgebraScalar 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]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) :=
Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
(zcGroupLike C G n.1 - 1) • x = 0Source 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 Lean proof
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]
Theorem
FoxDifferential.zcCompletedDifferentialModule_kernelAugmentationIdealMulStandard_smul_eq_zero
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) :=
Module.compHom _ (zcCompletedGroupAlgebraMap C hC psi)
(y : ZCCompletedGroupAlgebra C G) • x = 0The 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 Lean proof
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]