FoxDifferential.Completed.FreeProC.Uniqueness.SemidirectHom
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.
def freeProCZCCompletedFoxSemidirectHomOfCrossedDifferential
(ψ : F →* H) (delta : F → ZCFreeFoxCoordinates ProC.finiteQuotientClass (X := X) (H := H))
(hdelta : IsCrossedDifferential (zcCompletedGroupAlgebraScalar ProC.finiteQuotientClass ψ) delta) :
F →* ZCCompletedFoxSemidirect ProC.finiteQuotientClass X H where
toFun g := { left := delta g, right := ψ g }
map_one' := by
apply ZCCompletedFoxSemidirect.ext
· simpa using IsCrossedDifferential.one hdelta
· simp only [map_one, ZCCompletedFoxSemidirect.one_right]
map_mul' g h := by
apply ZCCompletedFoxSemidirect.ext
· simpa using hdelta g h
· simp only [map_mul, ZCCompletedFoxSemidirect.mul_right]A completed crossed differential and its coefficient homomorphism combine into a semidirect homomorphism.
theorem freeProCZCCompletedFoxSemidirectHomOfCrossedDifferential_left
(ψ : F →* H) (delta : F → ZCFreeFoxCoordinates ProC.finiteQuotientClass (X := X) (H := H))
(hdelta : IsCrossedDifferential (zcCompletedGroupAlgebraScalar ProC.finiteQuotientClass ψ) delta)
(g : F) :
(freeProCZCCompletedFoxSemidirectHomOfCrossedDifferential
(X := X) ψ delta hdelta g).left = delta gThe semidirect homomorphism attached to a crossed differential has \(\delta\) as its left component.
Show proof
rflProof. 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. 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. The finite-stage quotient maps preserve the chosen generators, so the crossed-derivation calculation does not depend on the representative of a coset. Linearity over the completed coefficient ring is checked after projection to each coefficient stage. The completed identity is therefore the unique compatible family whose coordinates are the verified finite Fox identities. 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 freeProCZCCompletedFoxSemidirectHomOfCrossedDifferential_right
(ψ : F →* H) (delta : F → ZCFreeFoxCoordinates ProC.finiteQuotientClass (X := X) (H := H))
(hdelta : IsCrossedDifferential (zcCompletedGroupAlgebraScalar ProC.finiteQuotientClass ψ) delta)
(g : F) :
(freeProCZCCompletedFoxSemidirectHomOfCrossedDifferential
(X := X) ψ delta hdelta g).right = ψ gThe semidirect homomorphism attached to a crossed differential has \(\psi\)s its right component.
Show proof
rflProof. 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. 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. The finite-stage quotient maps preserve the chosen generators, so the crossed-derivation calculation does not depend on the representative of a coset. Linearity over the completed coefficient ring is checked after projection to each coefficient stage. The completed identity is therefore the unique compatible family whose coordinates are the verified finite Fox identities. 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_freeProCZCCompletedFoxSemidirectGenerator_of_crossedDifferential
{ι : X → F} (hι : IsFreeProCGroup (ProC := ProC) ι)
(ψ : F →* H) (delta : F → ZCFreeFoxCoordinates ProC.finiteQuotientClass (X := X) (H := H))
(hdelta : IsCrossedDifferential (zcCompletedGroupAlgebraScalar ProC.finiteQuotientClass ψ) delta)
(hcontinuous :
Continuous (freeProCZCCompletedFoxSemidirectHomOfCrossedDifferential
(X := X) ψ delta hdelta))
(hbasis :
∀ x : X, delta (ι x) = Pi.single x (1 : ZCCompletedGroupAlgebra ProC.finiteQuotientClass H)) :
Continuous (freeProCZCCompletedFoxSemidirectGenerator (ProC := ProC) (fun x : X => ψ (ι x)))Show proof
by
have hgenerator :
(fun x : X =>
freeProCZCCompletedFoxSemidirectHomOfCrossedDifferential
(X := X) ψ delta hdelta (ι x)) =
freeProCZCCompletedFoxSemidirectGenerator (ProC := ProC) (fun x : X => ψ (ι x)) := by
funext x
apply ZCCompletedFoxSemidirect.ext
· exact hbasis x
· rfl
rw [← hgenerator]
exact hcontinuous.comp hι.continuous_ιProof. 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. 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. The finite-stage quotient maps preserve the chosen generators, so the crossed-derivation calculation does not depend on the representative of a coset. Linearity over the completed coefficient ring is checked after projection to each coefficient stage. The completed identity is therefore the unique compatible family whose coordinates are the verified finite Fox identities. 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.
□