FoxDifferential.Completed.DifferentialModule.TargetQuotient.Fundamental
This module develops the Fox-differential part of the theory. It records the formulas that connect generators, boundaries, Jacobians, and completed coordinates.
import
theorem ppCompletedGAFoxDerivToTarget_of_fundFormula_map
[Fintype X]
[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)]
(hfinite : ∀ a : ℕ,
Finite (FreeGroup X ⧸
finiteFoxCommutatorPowerSubgroup (F := FreeGroup X) N (ℓ ^ a)))
(w : FreeGroup X) :
primePowerCompletedGroupAlgebraMap
(ℓ := ℓ) (G := FreeGroup X)
(H := finiteFoxStageTargetQuotient (X := X) N)
(finiteFoxStageTargetQuotientContinuousMonoidHom (X := X) N)
(primePowerCompletedGroupAlgebraOf (ell := ℓ) (H := FreeGroup X) w) - 1 =
∑ i : X,
primePowerCompletedGroupAlgebraFreeFoxDerivativeToCompletedTarget
(ℓ := ℓ) (X := X) N hfinite i
(primePowerCompletedGroupAlgebraOf (ell := ℓ) (H := FreeGroup X) w) *
(primePowerCompletedGroupAlgebraMap
(ℓ := ℓ) (G := FreeGroup X)
(H := finiteFoxStageTargetQuotient (X := X) N)
(finiteFoxStageTargetQuotientContinuousMonoidHom (X := X) N)
(primePowerCompletedGroupAlgebraOf (ell := ℓ) (H := FreeGroup X)
(FreeGroup.of i)) - 1)The completed Fox derivative satisfies the fundamental formula after passage to the target quotient.
Show proof
by
rw [primePowerCompletedGroupAlgebraMap_targetQuotient_of]
simp_rw [primePowerCompletedGroupAlgebraMap_targetQuotient_of]
exact ppCompletedGAFoxDerivToTarget_of_fundFormula
(ℓ := ℓ) (X := X) N hfinite wProof. 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. 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. 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 ppCompletedGAFoxDerivToTarget_of_mul_product_map
[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)]
(hfinite : ∀ a : ℕ,
Finite (FreeGroup X ⧸
finiteFoxCommutatorPowerSubgroup (F := FreeGroup X) N (ℓ ^ a)))
(i : X) (u v : FreeGroup X) :
primePowerCompletedGroupAlgebraFreeFoxDerivativeToCompletedTarget
(ℓ := ℓ) (X := X) N hfinite i
(primePowerCompletedGroupAlgebraOf (ell := ℓ) (H := FreeGroup X) u *
primePowerCompletedGroupAlgebraOf (ell := ℓ) (H := FreeGroup X) v) =
primePowerCompletedGroupAlgebraFreeFoxDerivativeToCompletedTarget
(ℓ := ℓ) (X := X) N hfinite i
(primePowerCompletedGroupAlgebraOf (ell := ℓ) (H := FreeGroup X) u) +
primePowerCompletedGroupAlgebraMap
(ℓ := ℓ) (G := FreeGroup X)
(H := finiteFoxStageTargetQuotient (X := X) N)
(finiteFoxStageTargetQuotientContinuousMonoidHom (X := X) N)
(primePowerCompletedGroupAlgebraOf (ell := ℓ) (H := FreeGroup X) u) *
primePowerCompletedGroupAlgebraFreeFoxDerivativeToCompletedTarget
(ℓ := ℓ) (X := X) N hfinite i
(primePowerCompletedGroupAlgebraOf (ell := ℓ) (H := FreeGroup X) v)The completed Fox derivative of a product is computed by the crossed product rule after passage to the target quotient.
Show proof
by
rw [primePowerCompletedGroupAlgebraFreeFoxDerivativeToCompletedTarget_of_mul_product,
primePowerCompletedGroupAlgebraMap_targetQuotient_of]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 \(x x^{-1}=1\). 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 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. 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.
□