| import OAI.Algebra.AuslanderReiten.Ext.ExtKunneth |
| import OAI.Algebra.AuslanderReiten.HomComplex.HomCochainNaturality |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open CategoryTheory CategoryTheory.Abelian HomologicalComplex |
| open scoped ModuleCat.Algebra |
| namespace ArExplicit.ExternalTensor |
| universe u |
| variable (k R : Type u) [Field k] [Ring R] [Algebra k R] |
| attribute [local instance] moduleCatBaseModule moduleCatBaseTower |
| variable {X : ModuleCat.{u} R} (P : ProjectiveResolution X) |
| variable (Y : Type u) [AddCommGroup Y] [Module R Y] [Module k Y] [IsScalarTower k R Y] |
|
|
| theorem homCochain_mem_cycles (n : ℕ) (f : P.complex.X n →ₗ[R] Y) |
| (hf : f.comp (P.complex.d (n+1) n).hom=0) : |
| f ∈ LinearMap.ker ((homCochain k R P.complex Y).d n (n+1)).hom := by |
| change ((homCochain k R P.complex Y).d n (n+1)).hom f=0 |
| rw [homCochain_d] |
| exact hf |
|
|
| theorem homCochainNativeExtEquiv_literal (n : ℕ) |
| (f : P.complex.X n →ₗ[R] Y) (hf : f.comp (P.complex.d (n+1) n).hom=0) : |
| homCochainNativeExtEquiv k R P Y n |
| (VectorCochain.literalClass k (homCochain k R P.complex Y) n ⟨f,homCochain_mem_cycles k R P Y n f hf⟩) = |
| P.extMk (ModuleCat.ofHom (X := P.complex.X n) (Y := ModuleCat.of R Y) f) |
| (n+1) rfl (ModuleCat.hom_ext hf) := by |
| let C := canonicalHomCochain k R P.complex (ModuleCat.of R Y) |
| let z : LinearMap.ker (C.d n (n+1)).hom := ⟨f,by |
| change (C.d n (n+1)).hom f=0 |
| dsimp only [C, canonicalHomCochain] |
| rw [homCochain_d] |
| exact hf⟩ |
| have hz := VectorCochain.literalClass_map k (homCochainScalarIso k R P.complex Y).hom n |
| (⟨f,homCochain_mem_cycles k R P Y n f hf⟩ : LinearMap.ker ((homCochain k R P.complex Y).d n (n+1)).hom) z rfl |
| change homCochainExtEquiv k R (Y := ModuleCat.of R Y) P n |
| ((homologyMap (homCochainScalarIso k R P.complex Y).hom n).hom |
| (VectorCochain.literalClass k (homCochain k R P.complex Y) n ⟨f,homCochain_mem_cycles k R P Y n f hf⟩)) = _ |
| rw [hz] |
| have he := homCochainExtEquiv_cycleClass k R (Y := ModuleCat.of R Y) P n z |
| erw [homCochainCycleClass_eq_literal] at he |
| exact he |
|
|
| theorem homCochainExtEquiv_literal (N : ModuleCat.{u} R) (n : ℕ) |
| (f : P.complex.X n →ₗ[R] N) (hf : f.comp (P.complex.d (n+1) n).hom=0) : |
| homCochainExtEquiv k R (Y := N) P n |
| (VectorCochain.literalClass k (homCochain k R P.complex N) n |
| ⟨f,homCochain_mem_cycles k R P N n f hf⟩) = |
| P.extMk (ModuleCat.ofHom (X := P.complex.X n) (Y := N) f) |
| (n+1) rfl (ModuleCat.hom_ext hf) := by |
| let z : LinearMap.ker (homShort k R (Y := N) P n).g.hom := |
| ⟨f,homCochain_mem_cycles k R P N n f hf⟩ |
| have he := homCochainExtEquiv_cycleClass k R (Y := N) P n z |
| rw [homCochainCycleClass_eq_literal] at he |
| exact he |
|
|
| end ArExplicit.ExternalTensor |
|
|
| end |
|
|
| end OAI |
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|
