CompletedGroupAlgebra.Basic.AllFinite.Projections
This module supplies the topological part of the construction. It checks continuity and stagewise neighborhood properties so that the completed object inherits the required topology.
theorem continuous_completedGroupAlgebra_algebraMap :
Continuous (algebraMap R (Carrier R G))The coefficient-ring map \(R \to \widehat{R[G]}\) is continuous.
Show proof
by
letI : ∀ U : CompletedGroupAlgebraIndex G, TopologicalSpace (CompletedGroupAlgebraStage R G U) :=
fun U => (completedGroupAlgebraSystem R G).topologicalSpace U
have hval : Continuous fun r : R =>
fun U : CompletedGroupAlgebraIndex G =>
(show CompletedGroupAlgebraStage R G U from
(algebraMap R (Carrier R G) r).1 U) := by
change Continuous fun r : R =>
fun U : CompletedGroupAlgebraIndex G =>
algebraMap R (CompletedGroupAlgebraStage R G U) r
apply continuous_pi
intro U
exact finiteGroupAlgebra_algebraMap_continuous R (CompletedGroupAlgebraQuotient G U)
exact Continuous.subtype_mk hval fun r => (algebraMap R (Carrier R G) r).2Proof. 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. 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.
□def completedGroupAlgebraProjectionLinearMap (R : Type u) (G : Type v) [CommRing R]
[TopologicalSpace R] [IsTopologicalRing R] [Group G] [TopologicalSpace G]
[IsTopologicalGroup G] (U : CompletedGroupAlgebraIndex G) :
Carrier R G →ₗ[R] CompletedGroupAlgebraStage R G U where
toFun := completedGroupAlgebraProjection R G U
map_add' := completedGroupAlgebraProjection_add (R := R) (G := G) U
map_smul' := completedGroupAlgebraProjection_smul (R := R) (G := G) UThe canonical projection to a finite stage is bundled as an \(R\)-linear map.
def completedGroupAlgebraProjectionContinuousLinearMap (R : Type u) (G : Type v)
[CommRing R] [TopologicalSpace R] [IsTopologicalRing R] [Group G] [TopologicalSpace G]
[IsTopologicalGroup G] (U : CompletedGroupAlgebraIndex G) :
letI : TopologicalSpace (CompletedGroupAlgebraStage R G U) :=
(completedGroupAlgebraSystem R G).topologicalSpace U
Carrier R G →L[R] CompletedGroupAlgebraStage R G U := by
letI : TopologicalSpace (CompletedGroupAlgebraStage R G U) :=
(completedGroupAlgebraSystem R G).topologicalSpace U
exact
{ toLinearMap := completedGroupAlgebraProjectionLinearMap R G U
cont := (completedGroupAlgebraSystem R G).continuous_projection U }The finite-stage projection, as a continuous \(R\)-linear map.
def completedGroupAlgebraProjectionRingHom (R : Type u) (G : Type v) [CommRing R]
[TopologicalSpace R] [IsTopologicalRing R] [Group G] [TopologicalSpace G]
[IsTopologicalGroup G] (U : CompletedGroupAlgebraIndex G) :
Carrier R G →+* CompletedGroupAlgebraStage R G U where
toFun := completedGroupAlgebraProjection R G U
map_zero' := completedGroupAlgebraProjection_zero (R := R) (G := G) U
map_one' := completedGroupAlgebraProjection_one (R := R) (G := G) U
map_add' := completedGroupAlgebraProjection_add (R := R) (G := G) U
map_mul' := completedGroupAlgebraProjection_mul (R := R) (G := G) UThe canonical projection to a finite stage is bundled as a ring homomorphism.
def completedGroupAlgebraProjectionAlgHom (R : Type u) (G : Type v) [CommRing R]
[TopologicalSpace R] [IsTopologicalRing R] [Group G] [TopologicalSpace G]
[IsTopologicalGroup G] (U : CompletedGroupAlgebraIndex G) :
Carrier R G →ₐ[R] CompletedGroupAlgebraStage R G U where
toRingHom := completedGroupAlgebraProjectionRingHom R G U
commutes' := by
intro r
rflThe canonical projection to a finite stage is bundled as an \(R\)-algebra homomorphism.