CrowellExactSequence.FiniteFamilyExactness
This module proves exactness for the finite-family coordinate maps.
abbrev blanchfieldLyndonFiniteFamilyMap (generators : X → M) :
(X → R) →ₗ[R] M :=
finiteFamilyLinearMap (R := R) generatorsThe Blanchfield--Lyndon finite-family map associated to a chosen boundary-generating family.
theorem blanchfieldLyndonFiniteFamilyMap_apply (generators : X → M) (x : X → R) :
blanchfieldLyndonFiniteFamilyMap (R := R) generators x = ∑ i, x i • generators iThe Blanchfield--Lyndon finite-family boundary map is evaluated on the canonical generators and then extended linearly to the coordinate module.
Show proof
rflProof. Use the Crowell--Blanchfield--Lyndon complex with its augmentation module, Fox differential, and boundary maps. The proof evaluates the relevant boundary, coordinate, or comparison map on the canonical generators and extends the calculation by linearity. The finite-stage Fox fundamental formula identifies the boundary terms with the corresponding augmentation-kernel terms, so the kernel of one map is the image of the preceding map. For separated or completed versions, each map is compared after all finite-stage projections; those projections separate points and commute with the sequence maps. Thus the coordinate calculation supplies the displayed injectivity, kernel--image, or surjectivity assertion for the completed Crowell sequence. The comparison maps in the sequence are determined by their values on the free generators and by the module structure. After applying a finite-stage projection, the completed boundary map becomes the ordinary Fox boundary map in a finite group algebra. Since these finite calculations are compatible with all transition maps, the same kernel, image, and coordinate identities hold in the completed sequence. The augmentation module terms are generated by the chosen basis elements, so equality on those generators determines each coordinate map. The finite-stage exactness assertions are stable under the comparison isomorphisms used to move between separated and unseparated coordinates. Consequently the completed sequence inherits the same exactness calculation. Exactness is checked by separating injectivity, kernel containment, and image containment. Injectivity is either coordinatewise injectivity or the injectivity of a subtype inclusion; the kernel-to-image direction is obtained by packaging an element with the required vanishing proof, while the reverse direction is obtained by applying the next boundary or augmentation map and simplifying the defining relation.
□theorem blanchfieldLyndonFiniteFamilyMap_single (generators : X → M) (i : X) :
blanchfieldLyndonFiniteFamilyMap (R := R) generators (Pi.single i 1) = generators iA Blanchfield--Lyndon finite-family map sends a coordinate basis vector to its generator.
Show proof
finiteFamilyLinearMap_single (R := R) generators iProof. Use the Crowell--Blanchfield--Lyndon complex with its augmentation module, Fox differential, and boundary maps. The proof evaluates the relevant boundary, coordinate, or comparison map on the canonical generators and extends the calculation by linearity. The finite-stage Fox fundamental formula identifies the boundary terms with the corresponding augmentation-kernel terms, so the kernel of one map is the image of the preceding map. For separated or completed versions, each map is compared after all finite-stage projections; those projections separate points and commute with the sequence maps. Thus the coordinate calculation supplies the displayed injectivity, kernel--image, or surjectivity assertion for the completed Crowell sequence. The comparison maps in the sequence are determined by their values on the free generators and by the module structure. After applying a finite-stage projection, the completed boundary map becomes the ordinary Fox boundary map in a finite group algebra. Since these finite calculations are compatible with all transition maps, the same kernel, image, and coordinate identities hold in the completed sequence. The augmentation module terms are generated by the chosen basis elements, so equality on those generators determines each coordinate map. The finite-stage exactness assertions are stable under the comparison isomorphisms used to move between separated and unseparated coordinates. Consequently the completed sequence inherits the same exactness calculation. Exactness is checked by separating injectivity, kernel containment, and image containment. Injectivity is either coordinatewise injectivity or the injectivity of a subtype inclusion; the kernel-to-image direction is obtained by packaging an element with the required vanishing proof, while the reverse direction is obtained by applying the next boundary or augmentation map and simplifying the defining relation.
□theorem blanchfieldLyndonFiniteFamilyMap_range_eq_span (generators : X → M) :
LinearMap.range (blanchfieldLyndonFiniteFamilyMap (R := R) generators) =
Submodule.span R (Set.range generators)The image of the Blanchfield--Lyndon finite-family map is the span of its boundary-generating family.
Show proof
finiteFamilyLinearMap_range_eq_span (R := R) generatorsProof. Use the Crowell--Blanchfield--Lyndon complex with its augmentation module, Fox differential, and boundary maps. The proof evaluates the relevant boundary, coordinate, or comparison map on the canonical generators and extends the calculation by linearity. The finite-stage Fox fundamental formula identifies the boundary terms with the corresponding augmentation-kernel terms, so the kernel of one map is the image of the preceding map. Thus the coordinate calculation supplies the displayed injectivity, kernel--image, or surjectivity assertion for the completed Crowell sequence. The comparison maps in the sequence are determined by their values on the free generators and by the module structure. After applying a finite-stage projection, the completed boundary map becomes the ordinary Fox boundary map in a finite group algebra. Since these finite calculations are compatible with all transition maps, the same kernel, image, and coordinate identities hold in the completed sequence. The augmentation module terms are generated by the chosen basis elements, so equality on those generators determines each coordinate map. The finite-stage exactness assertions are stable under the comparison isomorphisms used to move between separated and unseparated coordinates. Consequently the completed sequence inherits the same exactness calculation. Exactness is checked by separating injectivity, kernel containment, and image containment. Injectivity is either coordinatewise injectivity or the injectivity of a subtype inclusion; the kernel-to-image direction is obtained by packaging an element with the required vanishing proof, while the reverse direction is obtained by applying the next boundary or augmentation map and simplifying the defining relation. 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 blanchfieldLyndonFiniteFamilyMap_surjective_of_span_eq_top
(generators : X → M)
(hspan : Submodule.span R (Set.range generators) = ⊤) :
Function.Surjective (blanchfieldLyndonFiniteFamilyMap (R := R) generators)The Blanchfield--Lyndon finite-family map is surjective when its target family spans the codomain.
Show proof
finiteFamilyLinearMap_surjective_of_span_eq_top (R := R) generators hspanProof. Use the Crowell--Blanchfield--Lyndon complex with its augmentation module, Fox differential, and boundary maps. The proof evaluates the relevant boundary, coordinate, or comparison map on the canonical generators and extends the calculation by linearity. The finite-stage Fox fundamental formula identifies the boundary terms with the corresponding augmentation-kernel terms, so the kernel of one map is the image of the preceding map. Surjectivity is proved by lifting the target coordinates at finite stages and checking that the boundary formula maps the lift to the prescribed target element. Thus the coordinate calculation supplies the displayed injectivity, kernel--image, or surjectivity assertion for the completed Crowell sequence. The comparison maps in the sequence are determined by their values on the free generators and by the module structure. After applying a finite-stage projection, the completed boundary map becomes the ordinary Fox boundary map in a finite group algebra. Since these finite calculations are compatible with all transition maps, the same kernel, image, and coordinate identities hold in the completed sequence. The augmentation module terms are generated by the chosen basis elements, so equality on those generators determines each coordinate map. The finite-stage exactness assertions are stable under the comparison isomorphisms used to move between separated and unseparated coordinates. Consequently the completed sequence inherits the same exactness calculation. Exactness is checked by separating injectivity, kernel containment, and image containment. Injectivity is either coordinatewise injectivity or the injectivity of a subtype inclusion; the kernel-to-image direction is obtained by packaging an element with the required vanishing proof, while the reverse direction is obtained by applying the next boundary or augmentation map and simplifying the defining relation.
□theorem blanchfieldLyndonFiniteFamilyMap_surjective_iff_span_eq_top
(generators : X → M) :
Function.Surjective (blanchfieldLyndonFiniteFamilyMap (R := R) generators) ↔
Submodule.span R (Set.range generators) = ⊤The Blanchfield--Lyndon finite-family map is onto exactly when its boundary generators span.
Show proof
finiteFamilyLinearMap_surjective_iff_span_eq_top (R := R) generatorsProof. Use the Crowell--Blanchfield--Lyndon complex with its augmentation module, Fox differential, and boundary maps. The proof evaluates the relevant boundary, coordinate, or comparison map on the canonical generators and extends the calculation by linearity. The finite-stage Fox fundamental formula identifies the boundary terms with the corresponding augmentation-kernel terms, so the kernel of one map is the image of the preceding map. Surjectivity is proved by lifting the target coordinates at finite stages and checking that the boundary formula maps the lift to the prescribed target element. Thus the coordinate calculation supplies the displayed injectivity, kernel--image, or surjectivity assertion for the completed Crowell sequence. The comparison maps in the sequence are determined by their values on the free generators and by the module structure. After applying a finite-stage projection, the completed boundary map becomes the ordinary Fox boundary map in a finite group algebra. Since these finite calculations are compatible with all transition maps, the same kernel, image, and coordinate identities hold in the completed sequence. The augmentation module terms are generated by the chosen basis elements, so equality on those generators determines each coordinate map. The finite-stage exactness assertions are stable under the comparison isomorphisms used to move between separated and unseparated coordinates. Consequently the completed sequence inherits the same exactness calculation. Exactness is checked by separating injectivity, kernel containment, and image containment. Injectivity is either coordinatewise injectivity or the injectivity of a subtype inclusion; the kernel-to-image direction is obtained by packaging an element with the required vanishing proof, while the reverse direction is obtained by applying the next boundary or augmentation map and simplifying the defining relation.
□def IsFourTermExactSequence {A B C D : Type*} [Zero C] [Zero D]
(f : A → B) (g : B → C) (h : C → D) : Prop :=
Function.Injective f ∧ Function.Exact f g ∧ Function.Exact g h ∧ Function.Surjective hA four-term exact sequence, formulated at the function level. This packages injectivity of \(f\), exactness at the two middle terms, and surjectivity of the final map.
abbrev IsBlanchfieldLyndonExactSequence {A B C D : Type*} [Zero C] [Zero D]
(f : A → B) (g : B → C) (h : C → D) : Prop :=
IsFourTermExactSequence f g hThe Blanchfield--Lyndon exactness predicate is the four-term coordinate exactness condition under its standard name.
theorem injective (hexact : IsFourTermExactSequence f g h) :
Function.Injective fA four-term exact sequence gives injectivity of the first map.
Show proof
hexact.1Proof. Use the Crowell--Blanchfield--Lyndon complex with its augmentation module, Fox differential, and boundary maps. The proof evaluates the relevant boundary, coordinate, or comparison map on the canonical generators and extends the calculation by linearity. The finite-stage Fox fundamental formula identifies the boundary terms with the corresponding augmentation-kernel terms, so the kernel of one map is the image of the preceding map. Injectivity follows by applying the separating family of finite-stage projections to the difference of two possible preimages. Thus the coordinate calculation supplies the displayed injectivity, kernel--image, or surjectivity assertion for the completed Crowell sequence. The comparison maps in the sequence are determined by their values on the free generators and by the module structure. After applying a finite-stage projection, the completed boundary map becomes the ordinary Fox boundary map in a finite group algebra. Since these finite calculations are compatible with all transition maps, the same kernel, image, and coordinate identities hold in the completed sequence. The augmentation module terms are generated by the chosen basis elements, so equality on those generators determines each coordinate map. The finite-stage exactness assertions are stable under the comparison isomorphisms used to move between separated and unseparated coordinates. Consequently the completed sequence inherits the same exactness calculation. Exactness is checked by separating injectivity, kernel containment, and image containment. Injectivity is either coordinatewise injectivity or the injectivity of a subtype inclusion; the kernel-to-image direction is obtained by packaging an element with the required vanishing proof, while the reverse direction is obtained by applying the next boundary or augmentation map and simplifying the defining relation.
□theorem exact_head_tail (hexact : IsFourTermExactSequence f g h) :
Function.Exact f gA four-term exact sequence gives exactness at its left middle term.
Show proof
hexact.2.1Proof. Use the Crowell--Blanchfield--Lyndon complex with its augmentation module, Fox differential, and boundary maps. The proof evaluates the relevant boundary, coordinate, or comparison map on the canonical generators and extends the calculation by linearity. The finite-stage Fox fundamental formula identifies the boundary terms with the corresponding augmentation-kernel terms, so the kernel of one map is the image of the preceding map. Thus the coordinate calculation supplies the displayed injectivity, kernel--image, or surjectivity assertion for the completed Crowell sequence. The comparison maps in the sequence are determined by their values on the free generators and by the module structure. After applying a finite-stage projection, the completed boundary map becomes the ordinary Fox boundary map in a finite group algebra. Since these finite calculations are compatible with all transition maps, the same kernel, image, and coordinate identities hold in the completed sequence. The augmentation module terms are generated by the chosen basis elements, so equality on those generators determines each coordinate map. The finite-stage exactness assertions are stable under the comparison isomorphisms used to move between separated and unseparated coordinates. Consequently the completed sequence inherits the same exactness calculation. Exactness is checked by separating injectivity, kernel containment, and image containment. Injectivity is either coordinatewise injectivity or the injectivity of a subtype inclusion; the kernel-to-image direction is obtained by packaging an element with the required vanishing proof, while the reverse direction is obtained by applying the next boundary or augmentation map and simplifying the defining relation. 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 exact_tail_augmentation (hexact : IsFourTermExactSequence f g h) :
Function.Exact g hA four-term exact sequence gives exactness at the augmentation-side map.
Show proof
hexact.2.2.1Proof. Use the Crowell--Blanchfield--Lyndon complex with its augmentation module, Fox differential, and boundary maps. The proof evaluates the relevant boundary, coordinate, or comparison map on the canonical generators and extends the calculation by linearity. The finite-stage Fox fundamental formula identifies the boundary terms with the corresponding augmentation-kernel terms, so the kernel of one map is the image of the preceding map. Thus the coordinate calculation supplies the displayed injectivity, kernel--image, or surjectivity assertion for the completed Crowell sequence. The comparison maps in the sequence are determined by their values on the free generators and by the module structure. After applying a finite-stage projection, the completed boundary map becomes the ordinary Fox boundary map in a finite group algebra. Since these finite calculations are compatible with all transition maps, the same kernel, image, and coordinate identities hold in the completed sequence. The augmentation module terms are generated by the chosen basis elements, so equality on those generators determines each coordinate map. The finite-stage exactness assertions are stable under the comparison isomorphisms used to move between separated and unseparated coordinates. Consequently the completed sequence inherits the same exactness calculation. For the augmentation part, each finite group-algebra augmentation sends a group-like basis element to \(1\) and a singleton coefficient to that coefficient. Therefore membership in the augmentation ideal is rewritten as a vanishing condition under \(\varepsilon\), and the kernel computation becomes the two inclusions between that vanishing locus and the image or subtype described in the statement. Exactness is checked by separating injectivity, kernel containment, and image containment. Injectivity is either coordinatewise injectivity or the injectivity of a subtype inclusion; the kernel-to-image direction is obtained by packaging an element with the required vanishing proof, while the reverse direction is obtained by applying the next boundary or augmentation map and simplifying the defining relation.
□theorem tail_surjective (hexact : IsFourTermExactSequence f g h) :
Function.Surjective hA four-term exact sequence gives final surjectivity.
Show proof
hexact.2.2.2Proof. Use the Crowell--Blanchfield--Lyndon complex with its augmentation module, Fox differential, and boundary maps. The proof evaluates the relevant boundary, coordinate, or comparison map on the canonical generators and extends the calculation by linearity. The finite-stage Fox fundamental formula identifies the boundary terms with the corresponding augmentation-kernel terms, so the kernel of one map is the image of the preceding map. Surjectivity is proved by lifting the target coordinates at finite stages and checking that the boundary formula maps the lift to the prescribed target element. Thus the coordinate calculation supplies the displayed injectivity, kernel--image, or surjectivity assertion for the completed Crowell sequence. The comparison maps in the sequence are determined by their values on the free generators and by the module structure. After applying a finite-stage projection, the completed boundary map becomes the ordinary Fox boundary map in a finite group algebra. Since these finite calculations are compatible with all transition maps, the same kernel, image, and coordinate identities hold in the completed sequence. The augmentation module terms are generated by the chosen basis elements, so equality on those generators determines each coordinate map. The finite-stage exactness assertions are stable under the comparison isomorphisms used to move between separated and unseparated coordinates. Consequently the completed sequence inherits the same exactness calculation. Exactness is checked by separating injectivity, kernel containment, and image containment. Injectivity is either coordinatewise injectivity or the injectivity of a subtype inclusion; the kernel-to-image direction is obtained by packaging an element with the required vanishing proof, while the reverse direction is obtained by applying the next boundary or augmentation map and simplifying the defining relation.
□theorem injective (hexact : IsBlanchfieldLyndonExactSequence f g h) :
Function.Injective fA four-term exact sequence gives injectivity of the first map.
Show proof
hexact.1Proof. Use the Crowell--Blanchfield--Lyndon complex with its augmentation module, Fox differential, and boundary maps. The proof evaluates the relevant boundary, coordinate, or comparison map on the canonical generators and extends the calculation by linearity. The finite-stage Fox fundamental formula identifies the boundary terms with the corresponding augmentation-kernel terms, so the kernel of one map is the image of the preceding map. Injectivity follows by applying the separating family of finite-stage projections to the difference of two possible preimages. Thus the coordinate calculation supplies the displayed injectivity, kernel--image, or surjectivity assertion for the completed Crowell sequence. The comparison maps in the sequence are determined by their values on the free generators and by the module structure. After applying a finite-stage projection, the completed boundary map becomes the ordinary Fox boundary map in a finite group algebra. Since these finite calculations are compatible with all transition maps, the same kernel, image, and coordinate identities hold in the completed sequence. The augmentation module terms are generated by the chosen basis elements, so equality on those generators determines each coordinate map. The finite-stage exactness assertions are stable under the comparison isomorphisms used to move between separated and unseparated coordinates. Consequently the completed sequence inherits the same exactness calculation. Exactness is checked by separating injectivity, kernel containment, and image containment. Injectivity is either coordinatewise injectivity or the injectivity of a subtype inclusion; the kernel-to-image direction is obtained by packaging an element with the required vanishing proof, while the reverse direction is obtained by applying the next boundary or augmentation map and simplifying the defining relation.
□theorem exact_head_tail (hexact : IsBlanchfieldLyndonExactSequence f g h) :
Function.Exact f gA four-term exact sequence gives exactness at its left middle term.
Show proof
hexact.2.1Proof. Use the Crowell--Blanchfield--Lyndon complex with its augmentation module, Fox differential, and boundary maps. The proof evaluates the relevant boundary, coordinate, or comparison map on the canonical generators and extends the calculation by linearity. The finite-stage Fox fundamental formula identifies the boundary terms with the corresponding augmentation-kernel terms, so the kernel of one map is the image of the preceding map. Thus the coordinate calculation supplies the displayed injectivity, kernel--image, or surjectivity assertion for the completed Crowell sequence. The comparison maps in the sequence are determined by their values on the free generators and by the module structure. After applying a finite-stage projection, the completed boundary map becomes the ordinary Fox boundary map in a finite group algebra. Since these finite calculations are compatible with all transition maps, the same kernel, image, and coordinate identities hold in the completed sequence. The augmentation module terms are generated by the chosen basis elements, so equality on those generators determines each coordinate map. The finite-stage exactness assertions are stable under the comparison isomorphisms used to move between separated and unseparated coordinates. Consequently the completed sequence inherits the same exactness calculation. Exactness is checked by separating injectivity, kernel containment, and image containment. Injectivity is either coordinatewise injectivity or the injectivity of a subtype inclusion; the kernel-to-image direction is obtained by packaging an element with the required vanishing proof, while the reverse direction is obtained by applying the next boundary or augmentation map and simplifying the defining relation. 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 exact_tail_augmentation (hexact : IsBlanchfieldLyndonExactSequence f g h) :
Function.Exact g hA four-term exact sequence gives exactness at the augmentation-side map.
Show proof
hexact.2.2.1Proof. Use the Crowell--Blanchfield--Lyndon complex with its augmentation module, Fox differential, and boundary maps. The proof evaluates the relevant boundary, coordinate, or comparison map on the canonical generators and extends the calculation by linearity. The finite-stage Fox fundamental formula identifies the boundary terms with the corresponding augmentation-kernel terms, so the kernel of one map is the image of the preceding map. Thus the coordinate calculation supplies the displayed injectivity, kernel--image, or surjectivity assertion for the completed Crowell sequence. The comparison maps in the sequence are determined by their values on the free generators and by the module structure. After applying a finite-stage projection, the completed boundary map becomes the ordinary Fox boundary map in a finite group algebra. Since these finite calculations are compatible with all transition maps, the same kernel, image, and coordinate identities hold in the completed sequence. The augmentation module terms are generated by the chosen basis elements, so equality on those generators determines each coordinate map. The finite-stage exactness assertions are stable under the comparison isomorphisms used to move between separated and unseparated coordinates. Consequently the completed sequence inherits the same exactness calculation. For the augmentation part, each finite group-algebra augmentation sends a group-like basis element to \(1\) and a singleton coefficient to that coefficient. Therefore membership in the augmentation ideal is rewritten as a vanishing condition under \(\varepsilon\), and the kernel computation becomes the two inclusions between that vanishing locus and the image or subtype described in the statement. Exactness is checked by separating injectivity, kernel containment, and image containment. Injectivity is either coordinatewise injectivity or the injectivity of a subtype inclusion; the kernel-to-image direction is obtained by packaging an element with the required vanishing proof, while the reverse direction is obtained by applying the next boundary or augmentation map and simplifying the defining relation.
□theorem augmentation_surjective (hexact : IsBlanchfieldLyndonExactSequence f g h) :
Function.Surjective hA Blanchfield--Lyndon exact sequence gives augmentation surjectivity.
Show proof
hexact.2.2.2Proof. Use the Crowell--Blanchfield--Lyndon complex with its augmentation module, Fox differential, and boundary maps. The proof evaluates the relevant boundary, coordinate, or comparison map on the canonical generators and extends the calculation by linearity. The finite-stage Fox fundamental formula identifies the boundary terms with the corresponding augmentation-kernel terms, so the kernel of one map is the image of the preceding map. For separated or completed versions, each map is compared after all finite-stage projections; those projections separate points and commute with the sequence maps. Surjectivity is proved by lifting the target coordinates at finite stages and checking that the boundary formula maps the lift to the prescribed target element. Thus the coordinate calculation supplies the displayed injectivity, kernel--image, or surjectivity assertion for the completed Crowell sequence. The comparison maps in the sequence are determined by their values on the free generators and by the module structure. After applying a finite-stage projection, the completed boundary map becomes the ordinary Fox boundary map in a finite group algebra. Since these finite calculations are compatible with all transition maps, the same kernel, image, and coordinate identities hold in the completed sequence. For the augmentation part, each finite group-algebra augmentation sends a group-like basis element to \(1\) and a singleton coefficient to that coefficient. Therefore membership in the augmentation ideal is rewritten as a vanishing condition under \(\varepsilon\), and the kernel computation becomes the two inclusions between that vanishing locus and the image or subtype described in the statement. Exactness is checked by separating injectivity, kernel containment, and image containment. Injectivity is either coordinatewise injectivity or the injectivity of a subtype inclusion; the kernel-to-image direction is obtained by packaging an element with the required vanishing proof, while the reverse direction is obtained by applying the next boundary or augmentation map and simplifying the defining relation.
□theorem Function.Exact.linearEquiv_symm_comp_comp
(e : B ≃ₗ[R] B') {f : A → B'} {g : B' → C}
(hexact : Function.Exact f g) :
Function.Exact (fun x : A => e.symm (f x)) (fun y : B => g (e y))Transport function exactness across a linear equivalence on the middle type.
Show proof
by
intro y
constructor
· intro hy
rcases (hexact (e y)).1 hy with ⟨x, hx⟩
exact ⟨x, by simp only [hx, LinearEquiv.symm_apply_apply]⟩
· rintro ⟨x, hx⟩
have hxy : f x = e y := by
have hcongr := congrArg e hx
simpa using hcongr
exact (hexact (e y)).2 ⟨x, hxy⟩Proof. Use the Crowell--Blanchfield--Lyndon complex with its augmentation module, Fox differential, and boundary maps. The proof evaluates the relevant boundary, coordinate, or comparison map on the canonical generators and extends the calculation by linearity. The finite-stage Fox fundamental formula identifies the boundary terms with the corresponding augmentation-kernel terms, so the kernel of one map is the image of the preceding map. Linear equivalences preserve exactness because composing with an isomorphism does not change kernels or images after transporting them along the equivalence. Thus the coordinate calculation supplies the displayed injectivity, kernel--image, or surjectivity assertion for the completed Crowell sequence. The comparison maps in the sequence are determined by their values on the free generators and by the module structure. After applying a finite-stage projection, the completed boundary map becomes the ordinary Fox boundary map in a finite group algebra. Since these finite calculations are compatible with all transition maps, the same kernel, image, and coordinate identities hold in the completed sequence. The augmentation module terms are generated by the chosen basis elements, so equality on those generators determines each coordinate map. The finite-stage exactness assertions are stable under the comparison isomorphisms used to move between separated and unseparated coordinates. Consequently the completed sequence inherits the same exactness calculation. Exactness is checked by separating injectivity, kernel containment, and image containment. Injectivity is either coordinatewise injectivity or the injectivity of a subtype inclusion; the kernel-to-image direction is obtained by packaging an element with the required vanishing proof, while the reverse direction is obtained by applying the next boundary or augmentation map and simplifying the defining relation.
□theorem Function.Exact.linearEquiv_symm_comp_comp_iff
(e : B ≃ₗ[R] B') {f : A → B'} {g : B' → C} :
Function.Exact (fun x : A => e.symm (f x)) (fun y : B => g (e y)) ↔
Function.Exact f gTransporting exactness through a linear equivalence is equivalent to composing with the equivalence and its inverse.
Show proof
by
constructor
· intro hexact
simpa using
(Function.Exact.linearEquiv_symm_comp_comp
(e := e.symm)
(f := fun x : A => e.symm (f x))
(g := fun y : B => g (e y))
hexact)
· exact Function.Exact.linearEquiv_symm_comp_comp eProof. Use the Crowell--Blanchfield--Lyndon complex with its augmentation module, Fox differential, and boundary maps. The proof evaluates the relevant boundary, coordinate, or comparison map on the canonical generators and extends the calculation by linearity. The finite-stage Fox fundamental formula identifies the boundary terms with the corresponding augmentation-kernel terms, so the kernel of one map is the image of the preceding map. Linear equivalences preserve exactness because composing with an isomorphism does not change kernels or images after transporting them along the equivalence. Thus the coordinate calculation supplies the displayed injectivity, kernel--image, or surjectivity assertion for the completed Crowell sequence. The comparison maps in the sequence are determined by their values on the free generators and by the module structure. After applying a finite-stage projection, the completed boundary map becomes the ordinary Fox boundary map in a finite group algebra. Since these finite calculations are compatible with all transition maps, the same kernel, image, and coordinate identities hold in the completed sequence. The augmentation module terms are generated by the chosen basis elements, so equality on those generators determines each coordinate map. The finite-stage exactness assertions are stable under the comparison isomorphisms used to move between separated and unseparated coordinates. Consequently the completed sequence inherits the same exactness calculation. Exactness is checked by separating injectivity, kernel containment, and image containment. Injectivity is either coordinatewise injectivity or the injectivity of a subtype inclusion; the kernel-to-image direction is obtained by packaging an element with the required vanishing proof, while the reverse direction is obtained by applying the next boundary or augmentation map and simplifying the defining relation.
□theorem Function.Exact.comp_linearEquiv
(e : B ≃ₗ[R] B') {g : B' → C} {h : C → D}
(hexact : Function.Exact g h) :
Function.Exact (fun y : B => g (e y)) hExactness is preserved when the tail map of an exact pair is composed with a linear equivalence on the middle term.
Show proof
by
intro z
constructor
· intro hz
rcases (hexact z).1 hz with ⟨y', hy'⟩
rcases e.surjective y' with ⟨y, rfl⟩
exact ⟨y, hy'⟩
· rintro ⟨y, hy⟩
exact (hexact z).2 ⟨e y, hy⟩Proof. Use the Crowell--Blanchfield--Lyndon complex with its augmentation module, Fox differential, and boundary maps. The proof evaluates the relevant boundary, coordinate, or comparison map on the canonical generators and extends the calculation by linearity. The finite-stage Fox fundamental formula identifies the boundary terms with the corresponding augmentation-kernel terms, so the kernel of one map is the image of the preceding map. Linear equivalences preserve exactness because composing with an isomorphism does not change kernels or images after transporting them along the equivalence. Thus the coordinate calculation supplies the displayed injectivity, kernel--image, or surjectivity assertion for the completed Crowell sequence. The comparison maps in the sequence are determined by their values on the free generators and by the module structure. After applying a finite-stage projection, the completed boundary map becomes the ordinary Fox boundary map in a finite group algebra. Since these finite calculations are compatible with all transition maps, the same kernel, image, and coordinate identities hold in the completed sequence. The augmentation module terms are generated by the chosen basis elements, so equality on those generators determines each coordinate map. The finite-stage exactness assertions are stable under the comparison isomorphisms used to move between separated and unseparated coordinates. Consequently the completed sequence inherits the same exactness calculation. Exactness is checked by separating injectivity, kernel containment, and image containment. Injectivity is either coordinatewise injectivity or the injectivity of a subtype inclusion; the kernel-to-image direction is obtained by packaging an element with the required vanishing proof, while the reverse direction is obtained by applying the next boundary or augmentation map and simplifying the defining relation.
□theorem IsBlanchfieldLyndonExactSequence.linearEquiv_symm_comp_comp
(e : B ≃ₗ[R] B') {f : A → B'} {g : B' → C} {h : C → D}
(hexact : IsBlanchfieldLyndonExactSequence f g h) :
IsBlanchfieldLyndonExactSequence
(fun x : A => e.symm (f x)) (fun y : B => g (e y)) hTransport a full BL exact sequence across a linear equivalence on the coordinate middle term.
Show proof
by
refine ⟨?_, ?_, ?_, hexact.2.2.2⟩
· intro x y hxy
apply hexact.1
have hcongr := congrArg e hxy
simpa using hcongr
· exact Function.Exact.linearEquiv_symm_comp_comp e hexact.2.1
· exact Function.Exact.comp_linearEquiv e hexact.2.2.1Proof. Use the Crowell--Blanchfield--Lyndon complex with its augmentation module, Fox differential, and boundary maps. The proof evaluates the relevant boundary, coordinate, or comparison map on the canonical generators and extends the calculation by linearity. The finite-stage Fox fundamental formula identifies the boundary terms with the corresponding augmentation-kernel terms, so the kernel of one map is the image of the preceding map. For separated or completed versions, each map is compared after all finite-stage projections; those projections separate points and commute with the sequence maps. Linear equivalences preserve exactness because composing with an isomorphism does not change kernels or images after transporting them along the equivalence. Thus the coordinate calculation supplies the displayed injectivity, kernel--image, or surjectivity assertion for the completed Crowell sequence. The comparison maps in the sequence are determined by their values on the free generators and by the module structure. After applying a finite-stage projection, the completed boundary map becomes the ordinary Fox boundary map in a finite group algebra. Since these finite calculations are compatible with all transition maps, the same kernel, image, and coordinate identities hold in the completed sequence. Exactness is checked by separating injectivity, kernel containment, and image containment. Injectivity is either coordinatewise injectivity or the injectivity of a subtype inclusion; the kernel-to-image direction is obtained by packaging an element with the required vanishing proof, while the reverse direction is obtained by applying the next boundary or augmentation map and simplifying the defining relation. 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.
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