FoxDifferential.Completed.DifferentialModule.TargetQuotient.Basic
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
def finiteFoxStageTargetQuotientContinuousMonoidHom
[TopologicalSpace (FreeGroup X)] [DiscreteTopology (FreeGroup X)]
(N : Subgroup (FreeGroup X)) [N.Normal]
[TopologicalSpace (finiteFoxStageTargetQuotient (X := X) N)] :
ContinuousMonoidHom (FreeGroup X) (finiteFoxStageTargetQuotient (X := X) N) where
toMonoidHom := QuotientGroup.mk' N
continuous_toFun := continuous_of_discreteTopologyThe completed Fox-differential map is continuous with respect to the inverse-limit topology on the completed coefficient modules.
theorem finiteFoxStageTargetQuotientContinuousMonoidHom_apply
[TopologicalSpace (FreeGroup X)] [DiscreteTopology (FreeGroup X)]
(N : Subgroup (FreeGroup X)) [N.Normal]
[TopologicalSpace (finiteFoxStageTargetQuotient (X := X) N)]
(w : FreeGroup X) :
finiteFoxStageTargetQuotientContinuousMonoidHom (X := X) N w =
QuotientGroup.mk' N wThe finite-stage target quotient homomorphism sends a word to its quotient class modulo \(N\).
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. 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. 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 finiteFoxStagePrimePowerSourceCompletedIndex_le_targetQuotientComap
[TopologicalSpace (FreeGroup X)]
[DiscreteTopology (FreeGroup X)]
(N : Subgroup (FreeGroup X)) [N.Normal]
[TopologicalSpace (finiteFoxStageTargetQuotient (X := X) N)]
[IsTopologicalGroup (finiteFoxStageTargetQuotient (X := X) N)]
(hfinite : ∀ a : ℕ,
Finite (FreeGroup X ⧸
finiteFoxCommutatorPowerSubgroup (F := FreeGroup X) N (ℓ ^ a)))
(j : PrimePowerCompletedGroupAlgebraIndex
(finiteFoxStageTargetQuotient (X := X) N)) :
completedGroupAlgebraComapIndex
(G := FreeGroup X) (H := finiteFoxStageTargetQuotient (X := X) N)
(finiteFoxStageTargetQuotientContinuousMonoidHom (X := X) N) j.2 ≤
finiteFoxStagePrimePowerSourceCompletedIndex
(ℓ := ℓ) (X := X) N hfinite j.1The completed free derivative source finite stage \([N,N]N^{ell^a}\) refines every finite stage pulled back from the target quotient \(F/N\). This order comparison lets the completed group-algebra projection for \(\pi: \mathbb{Z}_{\ell}\llbracket F\rrbracket \to \mathbb{Z}_{\ell}\llbracket F/N\rrbracket\) be computed from the completed free derivative source projection.
Show proof
by
change finiteFoxCommutatorPowerSubgroup (F := FreeGroup X) N (ℓ ^ j.1) ≤
Subgroup.comap
(finiteFoxStageTargetQuotientContinuousMonoidHom (X := X) N).toMonoidHom
(((OrderDual.ofDual j.2).1 :
OpenNormalSubgroup (finiteFoxStageTargetQuotient (X := X) N)) :
Subgroup (finiteFoxStageTargetQuotient (X := X) N))
intro g hg
change QuotientGroup.mk' N g ∈
(((OrderDual.ofDual j.2).1 :
OpenNormalSubgroup (finiteFoxStageTargetQuotient (X := X) N)) :
Subgroup (finiteFoxStageTargetQuotient (X := X) N))
have hgN : g ∈ N :=
finiteFoxCommutatorPowerSubgroup_le_normal (F := FreeGroup X) N (ℓ ^ j.1) hg
have hgq : QuotientGroup.mk' N g =
(1 : finiteFoxStageTargetQuotient (X := X) N) := by
simpa using (QuotientGroup.eq_one_iff (N := N) g).2 hgN
rw [hgq]
exact Subgroup.one_mem _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 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.
□