CompletedGroupAlgebra.InClassFunctoriality.GroupLike
Completed Group Algebra / Functoriality Within a Class / Group-Like.
theorem completedGroupAlgebraInClass_isCompletedGroupAlgebraModel
(C : ProCGroups.FiniteGroupClass.{v}) (hC : ProCGroups.FiniteGroupClass.FiniteOnly C)
(hForm : ProCGroups.FiniteGroupClass.Formation C)
(hR : IsProfiniteRing R) (hG : IsProCGroup C G) :
IsCompletedGroupAlgebraModel R G (CompletedGroupAlgebraInClass C hC R G)The \(C\)-indexed inverse-limit construction satisfies the universal-property specification for the completed group algebra when \(G\) is pro-\(C\).
Show proof
by
refine ⟨hR, hG.1, completedGroupAlgebraInClass_isProfiniteRing
(R := R) (G := G) C hC hR, ?_⟩
refine ⟨TopologicalSpace.induced (toCompletedGroupAlgebraInClass C hC R G)
(inferInstance : TopologicalSpace (CompletedGroupAlgebraInClass C hC R G)), ?_⟩
letI : TopologicalSpace (MonoidAlgebra R G) :=
TopologicalSpace.induced (toCompletedGroupAlgebraInClass C hC R G)
(inferInstance : TopologicalSpace (CompletedGroupAlgebraInClass C hC R G))
exact ⟨toCompletedGroupAlgebraInClassRingHom C hC R G,
denseRange_toCompletedGroupAlgebraInClass (R := R) (G := G) C hC hForm hG,
by
change Continuous (toCompletedGroupAlgebraInClass C hC R G)
exact continuous_induced_dom⟩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. 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.
□def completedGroupAlgebraOfInClass
(C : ProCGroups.FiniteGroupClass.{v}) (hC : ProCGroups.FiniteGroupClass.FiniteOnly C)
(R : Type u) (G : Type v) [CommRing R] [TopologicalSpace R] [IsTopologicalRing R]
[Group G] [TopologicalSpace G] [IsTopologicalGroup G]
(g : G) : CompletedGroupAlgebraInClass C hC R G :=
toCompletedGroupAlgebraInClass C hC R G (MonoidAlgebra.of R G g)A group element maps to its image in the \(C\)-indexed completed group algebra.
theorem completedGroupAlgebraProjectionInClass_of
(C : ProCGroups.FiniteGroupClass.{v}) (hC : ProCGroups.FiniteGroupClass.FiniteOnly C)
(U : CompletedGroupAlgebraIndexInClass G C) (g : G) :
completedGroupAlgebraProjectionInClass C hC R G U
(completedGroupAlgebraOfInClass C hC R G g) =
MonoidAlgebra.single (openNormalSubgroupInClassProj (C := C) (G := G) U g) 1Projection of a class-indexed completed group-like element to a finite quotient stage.
Show proof
by
rw [completedGroupAlgebraOfInClass,
completedGroupAlgebraProjectionInClass_toCompletedGroupAlgebraInClass,
completedGroupAlgebraStageMapInClass_of]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. 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. 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 completedGroupAlgebraOfInClass_one
(C : ProCGroups.FiniteGroupClass.{v}) (hC : ProCGroups.FiniteGroupClass.FiniteOnly C) :
completedGroupAlgebraOfInClass C hC R G 1 =
(1 : CompletedGroupAlgebraInClass C hC R G)The class-indexed completed group-like element attached to one is the unit.
Show proof
by
apply (completedGroupAlgebraSystemInClass C hC R G).ext
intro U
change completedGroupAlgebraProjectionInClass C hC R G U
(completedGroupAlgebraOfInClass C hC R G 1) =
completedGroupAlgebraProjectionInClass C hC R G U
(1 : CompletedGroupAlgebraInClass C hC R G)
rw [completedGroupAlgebraProjectionInClass_of]
change MonoidAlgebra.single (openNormalSubgroupInClassProj (C := C) (G := G) U 1)
(1 : R) =
completedGroupAlgebraProjectionRingHomInClass C hC R G U
(1 : CompletedGroupAlgebraInClass C hC R G)
rw [map_one]
change MonoidAlgebra.single (openNormalSubgroupInClassProj (C := C) (G := G) U 1)
(1 : R) = 1
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 completedGroupAlgebraOfInClass_mul
(C : ProCGroups.FiniteGroupClass.{v}) (hC : ProCGroups.FiniteGroupClass.FiniteOnly C)
(g h : G) :
completedGroupAlgebraOfInClass C hC R G (g * h) =
completedGroupAlgebraOfInClass C hC R G g *
completedGroupAlgebraOfInClass C hC R G hClass-indexed completed group-like elements multiply according to the group law.
Show proof
by
apply (completedGroupAlgebraSystemInClass C hC R G).ext
intro U
change completedGroupAlgebraProjectionInClass C hC R G U
(completedGroupAlgebraOfInClass C hC R G (g * h)) =
completedGroupAlgebraProjectionInClass C hC R G U
(completedGroupAlgebraOfInClass C hC R G g *
completedGroupAlgebraOfInClass C hC R G h)
rw [completedGroupAlgebraProjectionInClass_of]
have hmul :
completedGroupAlgebraProjectionInClass C hC R G U
(completedGroupAlgebraOfInClass C hC R G g *
completedGroupAlgebraOfInClass C hC R G h) =
completedGroupAlgebraProjectionInClass C hC R G U
(completedGroupAlgebraOfInClass C hC R G g) *
completedGroupAlgebraProjectionInClass C hC R G U
(completedGroupAlgebraOfInClass C hC R G h) := by
exact map_mul (completedGroupAlgebraProjectionRingHomInClass C hC R G U)
(completedGroupAlgebraOfInClass C hC R G g)
(completedGroupAlgebraOfInClass C hC R G h)
rw [hmul, completedGroupAlgebraProjectionInClass_of, completedGroupAlgebraProjectionInClass_of]
change MonoidAlgebra.single
(openNormalSubgroupInClassProj (C := C) (G := G) U (g * h)) (1 : R) =
MonoidAlgebra.single (openNormalSubgroupInClassProj (C := C) (G := G) U g) 1 *
MonoidAlgebra.single (openNormalSubgroupInClassProj (C := C) (G := G) U h) 1
simp only [map_mul, MonoidAlgebra.single_mul_single, mul_one]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. 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_completedGroupAlgebraStageMapInClass_of
(C : ProCGroups.FiniteGroupClass.{v}) (hC : ProCGroups.FiniteGroupClass.FiniteOnly C)
(U : CompletedGroupAlgebraIndexInClass G C) :
letI : TopologicalSpace (CompletedGroupAlgebraStageInClass C R G U)The class-indexed finite-stage group-like map is continuous.
Show proof
(completedGroupAlgebraSystemInClass C hC R G).topologicalSpace U
Continuous fun g : G => completedGroupAlgebraStageMapInClass C R G U
(MonoidAlgebra.of R G g) := by
letI : Finite (CompletedGroupAlgebraQuotientInClass G C U) :=
finite_completedGroupAlgebraQuotientInClass G C hC U
letI : TopologicalSpace (CompletedGroupAlgebraStageInClass C R G U) :=
(completedGroupAlgebraSystemInClass C hC R G).topologicalSpace U
letI : DiscreteTopology (CompletedGroupAlgebraQuotientInClass G C U) :=
QuotientGroup.discreteTopology
(ProCGroups.openNormalSubgroup_isOpen (G := G) ((OrderDual.ofDual U).1 : OpenNormalSubgroup G))
have hbasis :
Continuous fun q : CompletedGroupAlgebraQuotientInClass G C U =>
(MonoidAlgebra.of R (CompletedGroupAlgebraQuotientInClass G C U) q :
CompletedGroupAlgebraStageInClass C R G U) :=
continuous_of_discreteTopology
have hproj :
Continuous fun g : G => openNormalSubgroupInClassProj (C := C) (G := G) U g := by
change Continuous
(QuotientGroup.mk' (((OrderDual.ofDual U).1 : OpenNormalSubgroup G) : Subgroup G))
exact continuous_quotient_mk'
simpa [MonoidAlgebra.of, completedGroupAlgebraStageMapInClass_single] using hbasis.comp hprojProof. 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 continuous_completedGroupAlgebraOfInClass
(C : ProCGroups.FiniteGroupClass.{v}) (hC : ProCGroups.FiniteGroupClass.FiniteOnly C) :
Continuous (completedGroupAlgebraOfInClass C hC R G)The class-indexed completed group-like map is continuous.
Show proof
by
letI : ∀ U : CompletedGroupAlgebraIndexInClass G C,
TopologicalSpace (CompletedGroupAlgebraStageInClass C R G U) :=
fun U => (completedGroupAlgebraSystemInClass C hC R G).topologicalSpace U
have hval : Continuous fun g : G =>
fun U : CompletedGroupAlgebraIndexInClass G C =>
(show CompletedGroupAlgebraStageInClass C R G U from
(completedGroupAlgebraOfInClass C hC R G g).1 U) := by
change Continuous fun g : G =>
fun U : CompletedGroupAlgebraIndexInClass G C =>
completedGroupAlgebraStageMapInClass C R G U (MonoidAlgebra.of R G g)
apply continuous_pi
intro U
exact continuous_completedGroupAlgebraStageMapInClass_of (R := R) (G := G) C hC U
exact Continuous.subtype_mk hval fun g => (completedGroupAlgebraOfInClass C hC R G g).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. 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.
□