FoxDifferential.Completed.FreeProC.Uniqueness.Existence
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.
import
theorem existsUnique_freeProCZCCompletedFoxDerivativeVector_of_semidirectLiftHom
{ι : X → F} (hι : IsFreeProCGroup (ProC := ProC) ι)
(htarget : ProC (G := ZCCompletedFoxSemidirect ProC.finiteQuotientClass X H))
(φ : X → H)
(hφ : Continuous (freeProCZCCompletedFoxSemidirectGenerator (ProC := ProC) φ)) :
∃! delta : F → ZCFreeFoxCoordinates ProC.finiteQuotientClass (X := X) (H := H),
∃ f : F →ₜ* ZCCompletedFoxSemidirect ProC.finiteQuotientClass X H,
(∀ g : F, delta g = (f g).left) ∧
(∀ x : X, (f (ι x)).left =
Pi.single x (1 : ZCCompletedGroupAlgebra ProC.finiteQuotientClass H)) ∧
∀ x : X, (f (ι x)).right = φ xExistence and uniqueness of the free pro-\(C\) completed Fox derivative vector, formulated as the left component of a continuous semidirect homomorphism with prescribed generator data.
Show proof
by
refine ⟨freeProCZCCompletedFoxDerivativeVector
(ProC := ProC) hι htarget φ hφ, ?_, ?_⟩
· refine ⟨freeProCZCCompletedFoxSemidirectLiftHom
(ProC := ProC) hι htarget φ hφ, ?_, ?_, ?_⟩
· intro g
rfl
· exact freeProCZCCompletedFoxSemidirectLiftHom_left_generator
(ProC := ProC) hι htarget φ hφ
· exact freeProCZCCompletedFoxSemidirectLiftHom_right_generator
(ProC := ProC) hι htarget φ hφ
· intro delta hdelta
rcases hdelta with ⟨f, hdelta_left, hleft, hright⟩
have hf_eq := freeProCZCCompletedFoxSemidirectLiftHom_unique_of_components
(ProC := ProC) hι htarget φ hφ f hleft hright
funext g
calc
delta g = (f g).left := hdelta_left g
_ = (freeProCZCCompletedFoxSemidirectLiftHom
(ProC := ProC) hι htarget φ hφ g).left := by
rw [hf_eq]
_ = freeProCZCCompletedFoxDerivativeVector
(ProC := ProC) hι htarget φ hφ g := 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 \(x x^{-1}=1\). 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.
□theorem existsUnique_freeProCZCCompletedFoxSemidirectLift
{ι : X → F} (hι : IsFreeProCGroup (ProC := ProC) ι)
(htarget : ProC (G := ZCCompletedFoxSemidirect ProC.finiteQuotientClass X H))
(φ : X → H)
(hφ : Continuous (freeProCZCCompletedFoxSemidirectGenerator (ProC := ProC) φ)) :
∃! f : F →* ZCCompletedFoxSemidirect ProC.finiteQuotientClass X H,
Continuous f ∧
∀ x : X, f (ι x) = freeProCZCCompletedFoxSemidirectGenerator (ProC := ProC) φ xExistence and uniqueness of the continuous completed Fox semidirect lift from a free pro-\(C\) source.
Show proof
hι.existsUnique_lift htarget (freeProCZCCompletedFoxSemidirectGenerator (ProC := ProC) φ) hφProof. 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 \(x x^{-1}=1\). 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.
□theorem existsUnique_freeProCZCCompletedFoxSemidirectLift_components
{ι : X → F} (hι : IsFreeProCGroup (ProC := ProC) ι)
(htarget : ProC (G := ZCCompletedFoxSemidirect ProC.finiteQuotientClass X H))
(φ : X → H)
(hφ : Continuous (freeProCZCCompletedFoxSemidirectGenerator (ProC := ProC) φ)) :
∃! f : F →* ZCCompletedFoxSemidirect ProC.finiteQuotientClass X H,
Continuous f ∧
(∀ x : X, (f (ι x)).left =
Pi.single x (1 : ZCCompletedGroupAlgebra ProC.finiteQuotientClass H)) ∧
∀ x : X, (f (ι x)).right = φ xComponentwise existence and uniqueness of the continuous completed Fox semidirect lift from a free pro-\(C\) source.
Show proof
by
refine ⟨freeProCZCCompletedFoxSemidirectLift
(ProC := ProC) hι htarget φ hφ, ?_, ?_⟩
· exact ⟨
continuous_freeProCZCCompletedFoxSemidirectLift
(ProC := ProC) hι htarget φ hφ,
freeProCZCCompletedFoxSemidirectLift_left_generator
(ProC := ProC) hι htarget φ hφ,
freeProCZCCompletedFoxSemidirectLift_right_generator
(ProC := ProC) hι htarget φ hφ⟩
· intro f hf
exact freeProCZCCompletedFoxSemidirectLift_unique_of_components
(ProC := ProC) hι htarget φ hφ f hf.1 hf.2.1 hf.2.2Proof. 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 \(x x^{-1}=1\). 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.
□