CompletedGroupAlgebra.AllFiniteFunctoriality.Comap
This module develops the maps induced by continuous homomorphisms. It organizes the relevant quotient pullbacks and finite-stage maps, then proves the compatibility statements needed for the completed construction.
def completedGroupAlgebraComapIndex
(hG : ProCGroups.IsProfiniteGroup G) (φ : G →* H) (hφ : Continuous φ)
(V : CompletedGroupAlgebraIndex H) : CompletedGroupAlgebraIndex G :=
OrderDual.toDual
⟨ProCGroups.OpenNormalSubgroup.comap φ hφ ((OrderDual.ofDual V).1 : OpenNormalSubgroup H), by
letI : CompactSpace G := ProCGroups.IsProfiniteGroup.compactSpace hG
exact ProCGroups.openNormalSubgroup_finiteQuotient (G := G)
(ProCGroups.OpenNormalSubgroup.comap φ hφ ((OrderDual.ofDual V).1 : OpenNormalSubgroup H))⟩theorem completedGroupAlgebraComapIndex_subgroup
(hG : ProCGroups.IsProfiniteGroup G) (φ : G →* H) (hφ : Continuous φ)
(V : CompletedGroupAlgebraIndex H) :
(((OrderDual.ofDual (completedGroupAlgebraComapIndex (G := G) hG φ hφ V)).1 :
OpenNormalSubgroup G) : Subgroup G) =
(((OrderDual.ofDual V).1 : OpenNormalSubgroup H) : Subgroup H).comap φThe subgroup underlying the comap index is the subgroup-theoretic comap.
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, 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. 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. For quotient and subgroup claims, representatives are chosen in the ambient group and the induced map is checked to send the class of an element to the class of its image. Normality, openness, and membership in the finite quotient class are preserved by the subgroup, quotient, intersection, or inverse-image closure property being invoked.
□theorem completedGroupAlgebraComapIndex_mono
(hG : ProCGroups.IsProfiniteGroup G) (φ : G →* H) (hφ : Continuous φ)
{V W : CompletedGroupAlgebraIndex H} (hVW : V ≤ W) :
completedGroupAlgebraComapIndex (G := G) hG φ hφ V ≤
completedGroupAlgebraComapIndex (G := G) hG φ hφ WComap of all-finite completed-group-algebra indices is monotone.
Show proof
by
change (((OrderDual.ofDual W).1 : OpenNormalSubgroup H) : Subgroup H).comap φ ≤
(((OrderDual.ofDual V).1 : OpenNormalSubgroup H) : Subgroup H).comap φ
exact Subgroup.comap_mono hVWProof. 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. For quotient and subgroup claims, representatives are chosen in the ambient group and the induced map is checked to send the class of an element to the class of its image. Normality, openness, and membership in the finite quotient class are preserved by the subgroup, quotient, intersection, or inverse-image closure property being invoked.
□def completedGroupAlgebraComapQuotientMap
[IsTopologicalGroup H]
(hG : ProCGroups.IsProfiniteGroup G) (φ : G →* H) (hφ : Continuous φ)
(V : CompletedGroupAlgebraIndex H) :
CompletedGroupAlgebraQuotient G (completedGroupAlgebraComapIndex (G := G) hG φ hφ V) →*
CompletedGroupAlgebraQuotient H V :=
QuotientGroup.map _ _ φ (by
intro g hg
exact hg)The quotient homomorphism \(G/\varphi^{-1}(V) \to H/V\) induced by a continuous homomorphism \(\varphi: G \to H\).
theorem completedGroupAlgebraComapQuotientMap_mk
[IsTopologicalGroup H]
(hG : ProCGroups.IsProfiniteGroup G) (φ : G →* H) (hφ : Continuous φ)
(V : CompletedGroupAlgebraIndex H) (g : G) :
completedGroupAlgebraComapQuotientMap (G := G) hG φ hφ V
(QuotientGroup.mk'
((((OrderDual.ofDual (completedGroupAlgebraComapIndex (G := G) hG φ hφ V)).1 :
OpenNormalSubgroup G) : Subgroup G)) g) =
QuotientGroup.mk'
((((OrderDual.ofDual V).1 : OpenNormalSubgroup H) : Subgroup H)) (φ g)The quotient map associated with the inverse-image subgroup sends the class of \(g\in G\) to the class of \(\varphi(g)\in H\).
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, 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. 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. For quotient and subgroup claims, representatives are chosen in the ambient group and the induced map is checked to send the class of an element to the class of its image. Normality, openness, and membership in the finite quotient class are preserved by the subgroup, quotient, intersection, or inverse-image closure property being invoked.
□theorem completedGroupAlgebraComapQuotientMap_surjective
[IsTopologicalGroup H]
(hG : ProCGroups.IsProfiniteGroup G) (φ : G →* H) (hφ : Continuous φ)
(hφsurj : Function.Surjective φ) (V : CompletedGroupAlgebraIndex H) :
Function.Surjective (completedGroupAlgebraComapQuotientMap (G := G) hG φ hφ V)A surjective homomorphism induces a surjective map on the comap quotients.
Show proof
by
intro q
rcases QuotientGroup.mk'_surjective
((((OrderDual.ofDual V).1 : OpenNormalSubgroup H) : Subgroup H)) q with
⟨h, rfl⟩
rcases hφsurj h with ⟨g, rfl⟩
refine ⟨QuotientGroup.mk'
((((OrderDual.ofDual (completedGroupAlgebraComapIndex (G := G) hG φ hφ V)).1 :
OpenNormalSubgroup G) : Subgroup G)) g, ?_⟩
rw [completedGroupAlgebraComapQuotientMap_mk]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. Surjectivity is obtained by lifting a representative through the underlying surjective quotient or group homomorphism and then checking the defining finite-stage formula. 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. For surjectivity, choose a representative of the target coordinate and lift it through the underlying surjective group, quotient, or coefficient map. The defining formula for the induced map sends the constructed preimage to the chosen representative at every finite stage, so inverse-limit extensionality gives the required global preimage.
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