CompletedGroupAlgebra.Basic.AllFinite.Topology
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 completedGroupAlgebraStage_isTopologicalRing (U : CompletedGroupAlgebraIndex G) :
IsTopologicalRing ((completedGroupAlgebraSystem R G).X U)Each finite stage \(R[G/U]\) is a topological ring for its product topology.
Show proof
by
dsimp [completedGroupAlgebraSystem, CompletedGroupAlgebraStage]
exact finiteGroupAlgebra_isTopologicalRing R (CompletedGroupAlgebraQuotient G U)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. 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.
□instance instContinuousSMulCompletedGroupAlgebra :
ContinuousSMul R (Carrier R G) where
continuous_smul := by
let A := Carrier R G
letI : ∀ U : CompletedGroupAlgebraIndex G, TopologicalSpace (CompletedGroupAlgebraStage R G U) :=
fun U => (completedGroupAlgebraSystem R G).topologicalSpace U
have hval : Continuous fun p : R × A =>
fun U : CompletedGroupAlgebraIndex G =>
(show CompletedGroupAlgebraStage R G U from (p.1 • p.2).1 U) := by
change Continuous fun p : R × A =>
fun U : CompletedGroupAlgebraIndex G =>
p.1 • completedGroupAlgebraProjection R G U p.2
apply continuous_pi
intro U
letI : TopologicalSpace (CompletedGroupAlgebraStage R G U) :=
(completedGroupAlgebraSystem R G).topologicalSpace U
letI : ContinuousSMul R (CompletedGroupAlgebraStage R G U) :=
finiteGroupAlgebra_continuousSMul R (CompletedGroupAlgebraQuotient G U)
exact continuous_fst.smul
(((completedGroupAlgebraSystem R G).continuous_projection U).comp continuous_snd)
exact Continuous.subtype_mk hval fun p => (p.1 • p.2).2Scalar multiplication is continuous for the relevant inverse-limit topology.
instance instContinuousAddCompletedGroupAlgebra :
ContinuousAdd (Carrier R G) where
continuous_add := by
let A := Carrier R G
letI : ∀ U : CompletedGroupAlgebraIndex G, TopologicalSpace (CompletedGroupAlgebraStage R G U) :=
fun U => (completedGroupAlgebraSystem R G).topologicalSpace U
have hval : Continuous fun p : A × A =>
((p.1 + p.2 : A) :
(U : CompletedGroupAlgebraIndex G) → (completedGroupAlgebraSystem R G).X U) := by
change Continuous fun p : A × A =>
fun U : CompletedGroupAlgebraIndex G =>
completedGroupAlgebraProjection R G U p.1 +
completedGroupAlgebraProjection R G U p.2
apply continuous_pi
intro U
letI : IsTopologicalRing ((completedGroupAlgebraSystem R G).X U) :=
completedGroupAlgebraStage_isTopologicalRing (R := R) (G := G) U
exact (((completedGroupAlgebraSystem R G).continuous_projection U).comp continuous_fst).add
(((completedGroupAlgebraSystem R G).continuous_projection U).comp continuous_snd)
exact Continuous.subtype_mk hval fun p => (p.1 + p.2).2Addition on the all-finite completed group algebra is continuous for the inverse-limit topology.
instance instContinuousNegCompletedGroupAlgebra :
ContinuousNeg (Carrier R G) where
continuous_neg := by
let A := Carrier R G
letI : ∀ U : CompletedGroupAlgebraIndex G, TopologicalSpace (CompletedGroupAlgebraStage R G U) :=
fun U => (completedGroupAlgebraSystem R G).topologicalSpace U
have hval : Continuous fun x : A =>
((-x : A) :
(U : CompletedGroupAlgebraIndex G) → (completedGroupAlgebraSystem R G).X U) := by
change Continuous fun x : A =>
fun U : CompletedGroupAlgebraIndex G => -completedGroupAlgebraProjection R G U x
apply continuous_pi
intro U
letI : IsTopologicalRing ((completedGroupAlgebraSystem R G).X U) :=
completedGroupAlgebraStage_isTopologicalRing (R := R) (G := G) U
exact ((completedGroupAlgebraSystem R G).continuous_projection U).neg
exact Continuous.subtype_mk hval fun x => (-x).2Negation on the completed group algebra is continuous for the inverse-limit topology.
instance instContinuousMulCompletedGroupAlgebra :
ContinuousMul (Carrier R G) where
continuous_mul := by
let A := Carrier R G
letI : ∀ U : CompletedGroupAlgebraIndex G, TopologicalSpace (CompletedGroupAlgebraStage R G U) :=
fun U => (completedGroupAlgebraSystem R G).topologicalSpace U
have hval : Continuous fun p : A × A =>
((p.1 * p.2 : A) :
(U : CompletedGroupAlgebraIndex G) → (completedGroupAlgebraSystem R G).X U) := by
change Continuous fun p : A × A =>
fun U : CompletedGroupAlgebraIndex G =>
completedGroupAlgebraProjection R G U p.1 *
completedGroupAlgebraProjection R G U p.2
apply continuous_pi
intro U
letI : IsTopologicalRing ((completedGroupAlgebraSystem R G).X U) :=
completedGroupAlgebraStage_isTopologicalRing (R := R) (G := G) U
exact (((completedGroupAlgebraSystem R G).continuous_projection U).comp continuous_fst).mul
(((completedGroupAlgebraSystem R G).continuous_projection U).comp continuous_snd)
exact Continuous.subtype_mk hval fun p => (p.1 * p.2).2Multiplication on the completed group algebra is continuous for the inverse-limit topology.
instance instIsTopologicalRingCompletedGroupAlgebra :
IsTopologicalRing (Carrier R G) := by
letI : ContinuousAdd (Carrier R G) :=
instContinuousAddCompletedGroupAlgebra (R := R) (G := G)
letI : ContinuousMul (Carrier R G) :=
instContinuousMulCompletedGroupAlgebra (R := R) (G := G)
letI : ContinuousNeg (Carrier R G) :=
instContinuousNegCompletedGroupAlgebra (R := R) (G := G)
letI : IsTopologicalSemiring (Carrier R G) := IsTopologicalSemiring.mk
exact IsTopologicalRing.mkThe completed group algebra carries the canonical algebra structure over the coefficient ring.
theorem completedGroupAlgebraStage_isProfiniteRing
(hR : IsProfiniteRing R) (U : CompletedGroupAlgebraIndex G) :
IsProfiniteRing ((completedGroupAlgebraSystem R G).X U)Each finite stage is profinite when the coefficient ring is profinite.
Show proof
by
dsimp [completedGroupAlgebraSystem, CompletedGroupAlgebraStage]
exact finiteGroupAlgebra_isProfiniteRing R (CompletedGroupAlgebraQuotient G U) hRProof. 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. 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 completedGroupAlgebra_compactSpace (hR : IsProfiniteRing R) :
CompactSpace (Carrier R G)The completed group algebra is compact when the coefficient ring is profinite.
Show proof
by
letI : ∀ U : CompletedGroupAlgebraIndex G,
CompactSpace ((completedGroupAlgebraSystem R G).X U) := fun U =>
(completedGroupAlgebraStage_isProfiniteRing (R := R) (G := G) hR U).2.1
letI : ∀ U : CompletedGroupAlgebraIndex G,
T2Space ((completedGroupAlgebraSystem R G).X U) := fun U =>
(completedGroupAlgebraStage_isProfiniteRing (R := R) (G := G) hR U).2.2.1
infer_instanceProof. 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. 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 completedGroupAlgebra_t2Space (hR : IsProfiniteRing R) :
T2Space (Carrier R G)The completed group algebra is Hausdorff when the coefficient ring is profinite.
Show proof
by
letI : ∀ U : CompletedGroupAlgebraIndex G,
T2Space ((completedGroupAlgebraSystem R G).X U) := fun U =>
(completedGroupAlgebraStage_isProfiniteRing (R := R) (G := G) hR U).2.2.1
exact (completedGroupAlgebraSystem R G).t2Space_inverseLimitProof. 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. 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 completedGroupAlgebra_totallyDisconnectedSpace (hR : IsProfiniteRing R) :
TotallyDisconnectedSpace (Carrier R G)The completed group algebra is totally disconnected when the coefficient ring is profinite.
Show proof
by
letI : ∀ U : CompletedGroupAlgebraIndex G,
TotallyDisconnectedSpace ((completedGroupAlgebraSystem R G).X U) := fun U =>
(completedGroupAlgebraStage_isProfiniteRing (R := R) (G := G) hR U).2.2.2
exact (completedGroupAlgebraSystem R G).totallyDisconnectedSpace_inverseLimitProof. 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. 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 completedGroupAlgebra_isProfiniteRing (hR : IsProfiniteRing R) :
IsProfiniteRing (Carrier R G)The Section 5.3 completed group algebra is a profinite topological ring.
Show proof
by
letI : CompactSpace (Carrier R G) :=
completedGroupAlgebra_compactSpace (R := R) (G := G) hR
letI : T2Space (Carrier R G) :=
completedGroupAlgebra_t2Space (R := R) (G := G) hR
letI : TotallyDisconnectedSpace (Carrier R G) :=
completedGroupAlgebra_totallyDisconnectedSpace (R := R) (G := G) hR
exact ⟨inferInstance, inferInstance, inferInstance, inferInstance⟩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. 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 completedGroupAlgebra_isProfiniteModule (hR : IsProfiniteRing R) :
IsProfiniteModule R (Carrier R G)The completed group algebra is a profinite module over its profinite coefficient ring.
Show proof
by
letI : IsTopologicalRing R := hR.1
letI : IsTopologicalRing (Carrier R G) :=
instIsTopologicalRingCompletedGroupAlgebra (R := R) (G := G)
letI : IsTopologicalAddGroup (Carrier R G) := inferInstance
letI : ContinuousSMul R (Carrier R G) :=
instContinuousSMulCompletedGroupAlgebra (R := R) (G := G)
letI : CompactSpace (Carrier R G) :=
completedGroupAlgebra_compactSpace (R := R) (G := G) hR
letI : T2Space (Carrier R G) :=
completedGroupAlgebra_t2Space (R := R) (G := G) hR
letI : TotallyDisconnectedSpace (Carrier R G) :=
completedGroupAlgebra_totallyDisconnectedSpace (R := R) (G := G) hR
exact ⟨hR, inferInstance, inferInstance, inferInstance, inferInstance, inferInstance⟩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. 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 completedGroupAlgebraProjection_compatible
(x : Carrier R G) {U V : CompletedGroupAlgebraIndex G} (hUV : U ≤ V) :
completedGroupAlgebraTransition R G hUV (completedGroupAlgebraProjection R G V x) =
completedGroupAlgebraProjection R G U xThe finite-stage projection is compatible with the transition maps.
Show proof
(completedGroupAlgebraSystem R G).projection_compatible x U V hUVProof. 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. 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.
□