FoxDifferential.Completed.DifferentialModule.TargetQuotient.Surjective
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 finiteFoxStageTargetQuotientContinuousMonoidHom_surjective
[TopologicalSpace (FreeGroup X)] [DiscreteTopology (FreeGroup X)]
(N : Subgroup (FreeGroup X)) [N.Normal]
[TopologicalSpace (finiteFoxStageTargetQuotient (X := X) N)] :
Function.Surjective
(finiteFoxStageTargetQuotientContinuousMonoidHom (X := X) N)The quotient homomorphism \(\mathrm{FreeGroup}(X) \to \mathrm{FreeGroup}(X)/N\) is surjective. This specialized form is used to lift coefficients from \(\mathbb{Z}_{\ell}\llbracket F/N\rrbracket\) to \(\mathbb{Z}_{\ell}\llbracket F\rrbracket\) in the completed Fox derivative.
Show proof
by
intro q
rcases QuotientGroup.mk'_surjective N q with ⟨w, rfl⟩
exact ⟨w, rfl⟩Proof. Work from the defining Fox differential and its crossed-derivation rule. The values on generators determine the map; the product rule gives \(d(xy)=d(x)+x d(y)\), and the inverse rule follows by applying the product rule to \(x x^{-1}=1\). At finite coefficient or quotient stages, the source and target coordinates are obtained by applying the same quotient map to supports and the given coefficient map to coefficients. Surjectivity is obtained by choosing finite-stage lifts of the target coefficients and supports, verifying the derivative formula there, and assembling the compatible lifts. Completed assertions are checked after projection to every finite stage; continuity and closure follow from the inverse-limit topology and the closedness of the coordinate conditions. Finiteness at a stage follows because the quotient group and coefficient ring at that stage are finite, so the group-algebra support space is finite. Since the generator formulas determine the differential or homomorphism uniquely, equality of all finite-stage Fox coordinates proves the asserted completed identity. The finite-stage verification is made before passing to completion, so no topological density argument changes the algebraic calculation. Each completed coefficient system is obtained as a compatible family, and the quotient maps commute with the Fox differential by the generator formula. Thus the identity proved on generators persists under products, inverses, finite linear combinations, and finally under the inverse-limit passage. 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.
□theorem primePowerCompletedGroupAlgebraMap_targetQuotient_surjective
[TopologicalSpace (FreeGroup X)] [IsTopologicalGroup (FreeGroup X)]
[DiscreteTopology (FreeGroup X)]
(N : Subgroup (FreeGroup X)) [N.Normal]
[TopologicalSpace (finiteFoxStageTargetQuotient (X := X) N)]
[IsTopologicalGroup (finiteFoxStageTargetQuotient (X := X) N)] :
Function.Surjective
(primePowerCompletedGroupAlgebraMap
(ℓ := ℓ) (G := FreeGroup X)
(H := finiteFoxStageTargetQuotient (X := X) N)
(finiteFoxStageTargetQuotientContinuousMonoidHom (X := X) N))The completed group-algebra map attached to \(\mathrm{FreeGroup}(X) \to \mathrm{FreeGroup}(X)/N\) is surjective. This is the coefficient-lifting input for the surjectivity half of \(K/KI \to L\).
Show proof
by
exact
primePowerCompletedGroupAlgebraMap_surjective
(ℓ := ℓ) (G := FreeGroup X)
(H := finiteFoxStageTargetQuotient (X := X) N)
(finiteFoxStageTargetQuotientContinuousMonoidHom (X := X) N)
(finiteFoxStageTargetQuotientContinuousMonoidHom_surjective (X := X) N)Proof. Work from the defining Fox differential and its crossed-derivation rule. The values on generators determine the map; the product rule gives \(d(xy)=d(x)+x d(y)\), and the inverse rule follows by applying the product rule to \(x x^{-1}=1\). At finite coefficient or quotient stages, the source and target coordinates are obtained by applying the same quotient map to supports and the given coefficient map to coefficients. Surjectivity is obtained by choosing finite-stage lifts of the target coefficients and supports, verifying the derivative formula there, and assembling the compatible lifts. Completed assertions are checked after projection to every finite stage; continuity and closure follow from the inverse-limit topology and the closedness of the coordinate conditions. Finiteness at a stage follows because the quotient group and coefficient ring at that stage are finite, so the group-algebra support space is finite. Since the generator formulas determine the differential or homomorphism uniquely, equality of all finite-stage Fox coordinates proves the asserted completed identity. The finite-stage verification is made before passing to completion, so no topological density argument changes the algebraic calculation. Each completed coefficient system is obtained as a compatible family, and the quotient maps commute with the Fox differential by the generator formula. Thus the identity proved on generators persists under products, inverses, finite linear combinations, and finally under the inverse-limit passage. 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.
□def primePowerCompletedGroupAlgebraMap_targetQuotient_lift
[TopologicalSpace (FreeGroup X)] [IsTopologicalGroup (FreeGroup X)]
[DiscreteTopology (FreeGroup X)]
(N : Subgroup (FreeGroup X)) [N.Normal]
[TopologicalSpace (finiteFoxStageTargetQuotient (X := X) N)]
[IsTopologicalGroup (finiteFoxStageTargetQuotient (X := X) N)]
(a : PrimePowerCompletedGroupAlgebra ℓ (finiteFoxStageTargetQuotient (X := X) N)) :
PrimePowerCompletedGroupAlgebra ℓ (FreeGroup X) :=
Classical.choose
(primePowerCompletedGroupAlgebraMap_targetQuotient_surjective
(ℓ := ℓ) (X := X) N a)A noncomputable lift of a completed target group-algebra coefficient to the source completed group algebra. The defining equation is \(\mathrm{primePowerCompletedGroupAlgebraMap\_targetQuotient\_lift\_spec}\).
theorem primePowerCompletedGroupAlgebraMap_targetQuotient_lift_spec
[TopologicalSpace (FreeGroup X)] [IsTopologicalGroup (FreeGroup X)]
[DiscreteTopology (FreeGroup X)]
(N : Subgroup (FreeGroup X)) [N.Normal]
[TopologicalSpace (finiteFoxStageTargetQuotient (X := X) N)]
[IsTopologicalGroup (finiteFoxStageTargetQuotient (X := X) N)]
(a : PrimePowerCompletedGroupAlgebra ℓ (finiteFoxStageTargetQuotient (X := X) N)) :
primePowerCompletedGroupAlgebraMap
(ℓ := ℓ) (G := FreeGroup X)
(H := finiteFoxStageTargetQuotient (X := X) N)
(finiteFoxStageTargetQuotientContinuousMonoidHom (X := X) N)
(primePowerCompletedGroupAlgebraMap_targetQuotient_lift
(ℓ := ℓ) (X := X) N a) = aThe chosen coefficient lift maps back to the prescribed completed target coefficient.
Show proof
Classical.choose_spec
(primePowerCompletedGroupAlgebraMap_targetQuotient_surjective
(ℓ := ℓ) (X := X) N a)Proof. Work from the defining Fox differential and its crossed-derivation rule. The values on generators determine the map; the product rule gives \(d(xy)=d(x)+x d(y)\), and the inverse rule follows by applying the product rule to \(x x^{-1}=1\). At finite coefficient or quotient stages, the source and target coordinates are obtained by applying the same quotient map to supports and the given coefficient map to coefficients. Completed assertions are checked after projection to every finite stage; continuity and closure follow from the inverse-limit topology and the closedness of the coordinate conditions. Finiteness at a stage follows because the quotient group and coefficient ring at that stage are finite, so the group-algebra support space is finite. Since the generator formulas determine the differential or homomorphism uniquely, equality of all finite-stage Fox coordinates proves the asserted completed identity. The finite-stage verification is made before passing to completion, so no topological density argument changes the algebraic calculation. Each completed coefficient system is obtained as a compatible family, and the quotient maps commute with the Fox differential by the generator formula. Thus the identity proved on generators persists under products, inverses, finite linear combinations, and finally under the inverse-limit passage. 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. 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.
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