CompletedGroupAlgebra.AllFiniteFunctoriality.GroupLike
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.
theorem completedGroupAlgebraMap_of
(hG : ProCGroups.IsProfiniteGroup G) (φ : G →* H) (hφ : Continuous φ) (g : G) :
completedGroupAlgebraMap (G := G) (H := H) R hG φ hφ (completedGroupAlgebraOf R G g) =
completedGroupAlgebraOf R H (φ g)The completed functorial map sends the group-like element attached to \(g\) to that of \(\varphi(g)\).
Show proof
by
have h := congrFun
(congrArg DFunLike.coe
(completedGroupAlgebraMap_comp_toCompletedGroupAlgebra (R := R) (G := G) (H := H)
hG φ hφ))
(MonoidAlgebra.of R G g)
simpa [completedGroupAlgebraOf, finiteGroupAlgebra_mapDomainRingHom_of] using hProof. 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. 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. Coefficient and scalar compatibility is verified without changing the support in the finite quotient: only coefficients are transported by the given ring homomorphism or scalar action. Linearity, multiplicativity, and the algebra-map identities then extend the singleton computation to arbitrary finite sums. 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 completedGroupAlgebraMapAlgHom_of
(hG : ProCGroups.IsProfiniteGroup G) (φ : G →* H) (hφ : Continuous φ) (g : G) :
completedGroupAlgebraMapAlgHom (G := G) (H := H) R hG φ hφ
(completedGroupAlgebraOf R G g) =
completedGroupAlgebraOf R H (φ g)The algebra-hom completed functorial map sends group-like elements to their images.
Show proof
completedGroupAlgebraMap_of (R := R) (G := G) (H := H) hG φ hφ gProof. 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. 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. Coefficient and scalar compatibility is verified without changing the support in the finite quotient: only coefficients are transported by the given ring homomorphism or scalar action. Linearity, multiplicativity, and the algebra-map identities then extend the singleton computation to arbitrary finite sums. 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.
□