| import OAI.Algebra.AuslanderReiten.Ext.ExtKunnethMap |
| import OAI.Algebra.AuslanderReiten.HomComplex.HomCochainNativeClass |
| import OAI.Algebra.AuslanderReiten.Resolutions.MapResolutionExtConjugate |
|
|
| 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] |
| variable (H : homCochain k R P.complex Y ⟶ homCochain k R P.complex Y) |
|
|
| theorem nativeHomEndExt_extMk (n : ℕ) (f g : P.complex.X n →ₗ[R] Y) |
| (hf : f.comp (P.complex.d (n+1) n).hom = 0) |
| (hg : g.comp (P.complex.d (n+1) n).hom = 0) |
| (hfg : (H.f n).hom f = g) : |
| nativeHomEndExt k R P Y H n |
| (P.extMk (ModuleCat.ofHom (X := P.complex.X n) (Y := ModuleCat.of R Y) f) |
| (n+1) rfl (ModuleCat.hom_ext hf)) = |
| P.extMk (ModuleCat.ofHom (X := P.complex.X n) (Y := ModuleCat.of R Y) g) |
| (n+1) rfl (ModuleCat.hom_ext hg) := by |
| rw [← homCochainNativeExtEquiv_literal k R P Y n f hf, |
| ← homCochainNativeExtEquiv_literal k R P Y n g hg] |
| change (homCochainNativeExtEquiv k R P Y n) |
| ((homologyMap H n).hom ((homCochainNativeExtEquiv k R P Y n).symm |
| ((homCochainNativeExtEquiv k R P Y n) |
| (VectorCochain.literalClass k (homCochain k R P.complex Y) n _)))) = _ |
| rw [LinearEquiv.symm_apply_apply] |
| exact congrArg (homCochainNativeExtEquiv k R P Y n) |
| (VectorCochain.literalClass_map k H n _ _ hfg) |
|
|
| theorem nativeHomEnd_closed (n : ℕ) (f : P.complex.X n →ₗ[R] Y) |
| (hf : f.comp (P.complex.d (n+1) n).hom = 0) : |
| (show P.complex.X n →ₗ[R] Y from (H.f n).hom f).comp |
| (P.complex.d (n+1) n).hom = 0 := by |
| have hh := LinearMap.congr_fun (congrArg ModuleCat.Hom.hom (H.comm n (n+1))) f |
| change ((homCochain k R P.complex Y).d n (n+1)).hom ((H.f n).hom f) = |
| (H.f (n+1)).hom (((homCochain k R P.complex Y).d n (n+1)).hom f) at hh |
| rw [homCochain_d] at hh |
| change (show P.complex.X n →ₗ[R] Y from (H.f n).hom f).comp |
| (P.complex.d (n+1) n).hom = |
| (H.f (n+1)).hom (f.comp (P.complex.d (n+1) n).hom) at hh |
| rw [hf] at hh |
| exact hh.trans (H.f (n+1)).hom.map_zero |
|
|
| end ArExplicit.ExternalTensor |
|
|
| namespace ArExplicit.ExternalTensor |
| universe u |
| variable (k R : Type u) [Field k] [Ring R] [Algebra k R] |
| attribute [local instance] moduleCatBaseModule moduleCatBaseTower |
| variable (Y : Type u) [addY : AddCommGroup Y] [modY : Module R Y] |
| [baseY : Module k Y] [towerY : IsScalarTower k R Y] |
| variable (P : ProjectiveResolution (ModuleCat.of R Y)) |
| variable (H : homCochain k R P.complex Y ⟶ homCochain k R P.complex Y) |
| variable (F : ModuleCat.{u} R ⥤ ModuleCat.{u} R) [F.Additive] |
| [CategoryTheory.Limits.PreservesFiniteLimits F] |
| [CategoryTheory.Limits.PreservesFiniteColimits F] [F.PreservesProjectiveObjects] |
| variable (e : ModuleCat.of R Y ≅ F.obj (ModuleCat.of R Y)) |
| variable (φ : ProjectiveResolution.Hom P (ResolutionProducts.mapResolution F P) e.hom) |
|
|
| theorem nativeHomEndExt_eq_exactFunctor |
| (hφ : ∀ n (a : P.complex.X n ⟶ ModuleCat.of R Y), |
| ModuleCat.ofHom (X := P.complex.X n) (Y := ModuleCat.of R Y) ((H.f n).hom a.hom) = |
| (φ.hom.f n ≫ F.map a) ≫ e.inv) |
| (n : ℕ) (x : Ext (ModuleCat.of R Y) (ModuleCat.of R Y) n) : |
| nativeHomEndExt k R P Y H n x = |
| ((Ext.mk₀ e.hom).comp (x.mapExactFunctor F) (zero_add n)).comp |
| (Ext.mk₀ e.inv) (add_zero n) := by |
| obtain ⟨a, ha, rfl⟩ := P.extMk_surjective x (n+1) rfl |
| let b : P.complex.X n →ₗ[R] Y := (H.f n).hom a.hom |
| have ha' : a.hom.comp (P.complex.d (n+1) n).hom = 0 := |
| congrArg ModuleCat.Hom.hom ha |
| have hb := nativeHomEnd_closed k R P Y H n a.hom ha' |
| have ht := nativeHomEndExt_extMk k R P Y H n a.hom b ha' hb rfl |
| refine ht.trans ?_ |
| exact (ResolutionProducts.map_extMk_conjugate F P e.hom e.inv φ a ha |
| (ModuleCat.ofHom (X := P.complex.X n) (Y := ModuleCat.of R Y) b) |
| (ModuleCat.hom_ext hb) (hφ n a)).symm |
|
|
| end ArExplicit.ExternalTensor |
|
|
| end |
|
|
| end OAI |
|
|
