CompletedGroupAlgebra.InClassFunctoriality.UnitRepresentation
Completed Group Algebra / Functoriality Within a Class / Unit Representation.
theorem completedGroupAlgebraUnitRepresentationInClassConcrete_val
(C : ProCGroups.FiniteGroupClass.{v}) (hC : ProCGroups.FiniteGroupClass.FiniteOnly C)
(g : G) :
((completedGroupAlgebraUnitRepresentation R G (CompletedGroupAlgebraInClass C hC R G)
(toCompletedGroupAlgebraInClassRingHom C hC R G) g :
(CompletedGroupAlgebraInClass C hC R G)ˣ) : CompletedGroupAlgebraInClass C hC R G) =
completedGroupAlgebraOfInClass C hC R G gThe concrete class-indexed unit representation evaluates to the corresponding completed group-like element.
Show proof
rflProof. Let \(U\) range over the open normal finite-index subgroups of \(G\), and write the completed group algebra as the inverse limit \(\widehat{R[G]}=\varprojlim_U R[G/U]\). An element or morphism of the completed algebra is determined by its composites with all finite-stage projections, so the proof is reduced to ordinary group-algebra calculations at the stages \(R[G/U]\). At a finite stage, the coordinate operations are those of the ordinary group algebra; checking zero, addition, multiplication, scalar action, or evaluation after projection is therefore a direct coordinatewise calculation. Finiteness is reduced to the finite quotient set together with the finite coefficient data, so the relevant finite-stage function space or quotient object is finite. Compatibility under refinement of finite quotients ensures that the verified finite-stage identities form a compatible family, and inverse-limit extensionality then proves the claimed completed statement. Every equality used in the completed algebra is therefore an equality of compatible families. The finite-stage maps respect refinement because they are induced by the same quotient homomorphisms of groups and the same coefficient homomorphisms. This also handles definitions and bundled structures: the data are first built at each finite stage, then the compatibility equations show that the family lies in the inverse limit. Consequently the coordinate calculation uniquely determines the completed object and verifies the required algebraic or topological property. Functoriality and naturality are proved by comparing the two composites after projection to an arbitrary target stage. The source stage is the pullback or refined quotient required by the construction, and both composites are induced by the same quotient homomorphism together with the same coefficient map; hence the coordinate maps agree and so do the completed maps.
□theorem continuous_completedGroupAlgebraUnitRepresentationInClassConcrete_val
(C : ProCGroups.FiniteGroupClass.{v}) (hC : ProCGroups.FiniteGroupClass.FiniteOnly C) :
Continuous fun g : G =>
((completedGroupAlgebraUnitRepresentation R G (CompletedGroupAlgebraInClass C hC R G)
(toCompletedGroupAlgebraInClassRingHom C hC R G) g :
(CompletedGroupAlgebraInClass C hC R G)ˣ) : CompletedGroupAlgebraInClass C hC R G)The \(C\)-indexed canonical unit representation is continuous after forgetting to \(\widehat{R[G]}_{C}\).
Show proof
by
simpa using continuous_completedGroupAlgebraOfInClass (R := R) (G := G) C hCProof. Let \(U\) range over the open normal finite-index subgroups of \(G\), and write the completed group algebra as the inverse limit \(\widehat{R[G]}=\varprojlim_U R[G/U]\). An element or morphism of the completed algebra is determined by its composites with all finite-stage projections, so the proof is reduced to ordinary group-algebra calculations at the stages \(R[G/U]\). At a finite stage, quotient maps move the support from one coset space to another, while coefficient maps act only on coefficients; therefore the relevant formula is checked on singleton basis functions and then extended by additivity and multiplicativity. Finiteness is reduced to the finite quotient set together with the finite coefficient data, so the relevant finite-stage function space or quotient object is finite. Continuity is checked by the initial topology of the inverse limit: after composing with every finite-stage projection, the resulting map is a finite-stage homomorphism or a composition of such homomorphisms. Compatibility under refinement of finite quotients ensures that the verified finite-stage identities form a compatible family, and inverse-limit extensionality then proves the claimed completed statement. Every equality used in the completed algebra is therefore an equality of compatible families. The finite-stage maps respect refinement because they are induced by the same quotient homomorphisms of groups and the same coefficient homomorphisms. This also handles definitions and bundled structures: the data are first built at each finite stage, then the compatibility equations show that the family lies in the inverse limit. Consequently the coordinate calculation uniquely determines the completed object and verifies the required algebraic or topological property. The topological assertion is checked by the initial topology of the inverse limit. After composing with each finite-stage projection, the relevant map is a continuous finite-stage homomorphism or an operation on a finite product; compactness, Hausdorffness, total disconnectedness, and profiniteness are then inherited from the finite stages by the standard inverse-limit argument.
□theorem toCompletedGroupAlgebraInClassRingHom_mem_span_completedGroupAlgebraOfInClass
(C : ProCGroups.FiniteGroupClass.{v}) (hC : ProCGroups.FiniteGroupClass.FiniteOnly C)
(x : MonoidAlgebra R G) :
toCompletedGroupAlgebraInClassRingHom C hC R G x ∈
Submodule.span R (Set.range (completedGroupAlgebraOfInClass C hC R G))The dense abstract map lands in the span of class-indexed completed group-like elements.
Show proof
by
classical
induction x using Finsupp.induction with
| zero =>
rw [show toCompletedGroupAlgebraInClassRingHom C hC R G (0 : MonoidAlgebra R G) =
(0 : CompletedGroupAlgebraInClass C hC R G) by
exact map_zero (toCompletedGroupAlgebraInClassRingHom C hC R G)]
exact Submodule.zero_mem _
| single_add g r x _ _ ih =>
rw [map_add]
refine Submodule.add_mem _ ?_ ih
have hsingle :
toCompletedGroupAlgebraInClassRingHom C hC R G (MonoidAlgebra.single g r) =
r • completedGroupAlgebraOfInClass C hC R G g := by
rw [show MonoidAlgebra.single g r =
r • MonoidAlgebra.of R G g by
simp only [MonoidAlgebra.of, MonoidHom.coe_mk, OneHom.coe_mk, MonoidAlgebra.smul_single, smul_eq_mul, mul_one]]
change toCompletedGroupAlgebraInClass C hC R G (r • MonoidAlgebra.of R G g) =
r • completedGroupAlgebraOfInClass C hC R G g
rw [toCompletedGroupAlgebraInClass_smul]
rfl
rw [hsingle]
exact Submodule.smul_mem _ r (Submodule.subset_span ⟨g, rfl⟩)Proof. Let \(U\) range over the open normal finite-index subgroups of \(G\), and write the completed group algebra as the inverse limit \(\widehat{R[G]}=\varprojlim_U R[G/U]\). An element or morphism of the completed algebra is determined by its composites with all finite-stage projections, so the proof is reduced to ordinary group-algebra calculations at the stages \(R[G/U]\). At a finite stage, quotient maps move the support from one coset space to another, while coefficient maps act only on coefficients; therefore the relevant formula is checked on singleton basis functions and then extended by additivity and multiplicativity. Kernel, image, and exactness assertions are proved by the two standard inclusions: an image element satisfies the required vanishing condition, and an element satisfying that condition is repackaged with the vanishing proof as a preimage. Finiteness is reduced to the finite quotient set together with the finite coefficient data, so the relevant finite-stage function space or quotient object is finite. Continuity is checked by the initial topology of the inverse limit: after composing with every finite-stage projection, the resulting map is a finite-stage homomorphism or a composition of such homomorphisms. For dense-image and closure arguments, the equality is first established on the abstract group algebra and then passed to the completed object using finite-stage separation and closedness in the profinite topology. Compatibility under refinement of finite quotients ensures that the verified finite-stage identities form a compatible family, and inverse-limit extensionality then proves the claimed completed statement. Functoriality and naturality are proved by comparing the two composites after projection to an arbitrary target stage. The source stage is the pullback or refined quotient required by the construction, and both composites are induced by the same quotient homomorphism together with the same coefficient map; hence the coordinate maps agree and so do the completed maps.
□theorem completedGroupAlgebraOfInClass_dense_span
(C : ProCGroups.FiniteGroupClass.{v}) (hC : ProCGroups.FiniteGroupClass.FiniteOnly C)
(hForm : ProCGroups.FiniteGroupClass.Formation C) (hG : IsProCGroup C G) :
closure (Submodule.span R (Set.range (completedGroupAlgebraOfInClass C hC R G)) :
Set (CompletedGroupAlgebraInClass C hC R G)) = Set.univThe \(C\)-indexed completed group-like elements topologically generate \(\widehat{R[G]}_{C}\) as an \(R\)-module, for pro-\(C\) groups and formation classes.
Show proof
by
rw [Set.eq_univ_iff_forall]
intro y
have hy :
y ∈ closure (Set.range (toCompletedGroupAlgebraInClassRingHom C hC R G)) := by
have hdense : DenseRange (toCompletedGroupAlgebraInClassRingHom C hC R G) := by
change DenseRange (toCompletedGroupAlgebraInClass C hC R G)
exact denseRange_toCompletedGroupAlgebraInClass (R := R) (G := G) C hC hForm hG
rw [hdense.closure_range]
exact Set.mem_univ y
exact closure_mono (by
intro z hz
rcases hz with ⟨x, rfl⟩
exact toCompletedGroupAlgebraInClassRingHom_mem_span_completedGroupAlgebraOfInClass
(R := R) (G := G) C hC x) hyProof. Let \(U\) range over the open normal finite-index subgroups of \(G\), and write the completed group algebra as the inverse limit \(\widehat{R[G]}=\varprojlim_U R[G/U]\). An element or morphism of the completed algebra is determined by its composites with all finite-stage projections, so the proof is reduced to ordinary group-algebra calculations at the stages \(R[G/U]\). At a finite stage, the coordinate operations are those of the ordinary group algebra; checking zero, addition, multiplication, scalar action, or evaluation after projection is therefore a direct coordinatewise calculation. Kernel, image, and exactness assertions are proved by the two standard inclusions: an image element satisfies the required vanishing condition, and an element satisfying that condition is repackaged with the vanishing proof as a preimage. Finiteness is reduced to the finite quotient set together with the finite coefficient data, so the relevant finite-stage function space or quotient object is finite. Continuity is checked by the initial topology of the inverse limit: after composing with every finite-stage projection, the resulting map is a finite-stage homomorphism or a composition of such homomorphisms. For dense-image and closure arguments, the equality is first established on the abstract group algebra and then passed to the completed object using finite-stage separation and closedness in the profinite topology. Compatibility under refinement of finite quotients ensures that the verified finite-stage identities form a compatible family, and inverse-limit extensionality then proves the claimed completed statement. The topological assertion is checked by the initial topology of the inverse limit. After composing with each finite-stage projection, the relevant map is a continuous finite-stage homomorphism or an operation on a finite product; compactness, Hausdorffness, total disconnectedness, and profiniteness are then inherited from the finite stages by the standard inverse-limit argument.
□theorem completedGroupAlgebraInClass_module_induces_continuous_gmodule
(C : ProCGroups.FiniteGroupClass.{v}) (hC : ProCGroups.FiniteGroupClass.FiniteOnly C)
(A : Type (max u v)) [AddCommGroup A] [TopologicalSpace A]
[Module (CompletedGroupAlgebraInClass C hC R G) A]
[ContinuousSMul (CompletedGroupAlgebraInClass C hC R G) A] :
letI : DistribMulAction G AA continuous module over the \(C\)-indexed completed group algebra inherits the natural continuous \(G\)-module structure via the group-like units.
Show proof
unitRepresentationDistribMulAction G (CompletedGroupAlgebraInClass C hC R G) A
(completedGroupAlgebraUnitRepresentation R G (CompletedGroupAlgebraInClass C hC R G) (toCompletedGroupAlgebraInClassRingHom C hC R G))
ContinuousSMul G A := by
exact unitRepresentation_continuousSMul G (CompletedGroupAlgebraInClass C hC R G) A
(completedGroupAlgebraUnitRepresentation R G (CompletedGroupAlgebraInClass C hC R G) (toCompletedGroupAlgebraInClassRingHom C hC R G))
(continuous_completedGroupAlgebraUnitRepresentationInClassConcrete_val (R := R) (G := G) C hC)Proof. Let \(U\) range over the open normal finite-index subgroups of \(G\), and write the completed group algebra as the inverse limit \(\widehat{R[G]}=\varprojlim_U R[G/U]\). An element or morphism of the completed algebra is determined by its composites with all finite-stage projections, so the proof is reduced to ordinary group-algebra calculations at the stages \(R[G/U]\). At a finite stage, the coordinate operations are those of the ordinary group algebra; checking zero, addition, multiplication, scalar action, or evaluation after projection is therefore a direct coordinatewise calculation. Coefficient-change assertions hold because each singleton coefficient is transported by the chosen ring homomorphism and the quotient support is left unchanged. Finiteness is reduced to the finite quotient set together with the finite coefficient data, so the relevant finite-stage function space or quotient object is finite. Continuity is checked by the initial topology of the inverse limit: after composing with every finite-stage projection, the resulting map is a finite-stage homomorphism or a composition of such homomorphisms. Compatibility under refinement of finite quotients ensures that the verified finite-stage identities form a compatible family, and inverse-limit extensionality then proves the claimed completed statement. The topological assertion is checked by the initial topology of the inverse limit. After composing with each finite-stage projection, the relevant map is a continuous finite-stage homomorphism or an operation on a finite product; compactness, Hausdorffness, total disconnectedness, and profiniteness are then inherited from the finite stages by the standard inverse-limit argument. Functoriality and naturality are proved by comparing the two composites after projection to an arbitrary target stage. The source stage is the pullback or refined quotient required by the construction, and both composites are induced by the same quotient homomorphism together with the same coefficient map; hence the coordinate maps agree and so do the completed maps.
□