| import OAI.Algebra.AuslanderReiten.Ext.ExtKunnethClass |
| import OAI.Algebra.AuslanderReiten.Tensor.ExternalTensorCubicExt |
| import OAI.Algebra.AuslanderReiten.Ext.TensorExtCoefficients |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open CategoryTheory CategoryTheory.Abelian |
| open scoped TensorProduct ModuleCat.Algebra |
| namespace ArExplicit.ExternalTensor |
| universe u |
| variable (k A B : Type u) [Field k] [Ring A] [Ring B] [Algebra k A] [Algebra k B] |
| variable [CharP k 2] [Module.Finite k A] [Module.Finite k B] |
| variable [HasDerivedCategory (ModuleCat.{u} A)] [HasDerivedCategory (ModuleCat.{u} B)] |
| variable [HasDerivedCategory (ModuleCat.{u} (A ⊗[k] B))] |
| attribute [local instance] moduleCatBaseModule moduleCatBaseTower |
| variable {X : ModuleCat.{u} A} {Y : ModuleCat.{u} B} |
| variable (P : ProjectiveResolution X) (Q : ProjectiveResolution Y) |
| variable [∀ n, Module.Finite A (P.complex.X n)] [∀ n, Module.Finite B (Q.complex.X n)] |
| variable (U : ModuleCat.{u} A) (V : ModuleCat.{u} B) |
|
|
| private theorem extMk_of_eq {R : Type u} [Ring R] {M N : ModuleCat.{u} R} |
| (L : ProjectiveResolution M) {n : ℕ} (f g : L.complex.X n ⟶ N) |
| (hf : L.complex.d (n+1) n ≫ f=0) (hg : L.complex.d (n+1) n ≫ g=0) (hfg : f=g) : |
| L.extMk f (n+1) rfl hf = L.extMk g (n+1) rfl hg := by |
| subst g |
| rfl |
|
|
| def elementaryExt (i j n : ℕ) (h : i+j=n) |
| (f : P.complex.X i →ₗ[A] U) (g : Q.complex.X j →ₗ[B] V) |
| (hf : f.comp (P.complex.d (i+1) i).hom=0) (hg : g.comp (Q.complex.d (j+1) j).hom=0) : |
| Ext (moduleObject k A B X Y) (moduleObject k A B U V) n := |
| (projectiveResolution k A B P Q).extMk |
| (ModuleCat.ofHom (X := (complex k A B P.complex Q.complex).X n) |
| (Y := moduleObject k A B U V) (elementaryHom k A B P.complex Q.complex U V i j n h f g)) |
| (n+1) rfl (ModuleCat.hom_ext (elementary_closed k A B P.complex Q.complex U V i j n h f g hf hg)) |
|
|
| omit [CharP k 2] [Module.Finite k A] [Module.Finite k B] |
| [HasDerivedCategory (ModuleCat.{u} A)] [HasDerivedCategory (ModuleCat.{u} B)] |
| [HasDerivedCategory (ModuleCat.{u} (A ⊗[k] B))] in |
| theorem elementaryExt_kunneth (i j n : ℕ) (h : i+j=n) |
| (f : P.complex.X i →ₗ[A] U) (g : Q.complex.X j →ₗ[B] V) |
| (hf : f.comp (P.complex.d (i+1) i).hom=0) (hg : g.comp (Q.complex.d (j+1) j).hom=0) : |
| extKunnethEquiv k A B P Q U V n (elementaryExt k A B P Q U V i j n h f g hf hg) = |
| (Pi.single ⟨(i,j),h⟩ |
| (P.extMk (ModuleCat.ofHom (X := P.complex.X i) (Y := U) f) (i+1) rfl (ModuleCat.hom_ext hf) ⊗ₜ[k] |
| Q.extMk (ModuleCat.ofHom (X := Q.complex.X j) (Y := V) g) (j+1) rfl (ModuleCat.hom_ext hg)) : |
| extCoefficients k A B X U Y V n) := |
| extKunnethEquiv_elementary k A B P Q U V i j n h f g hf hg |
|
|
| variable (p : ∀ n, P.complex.X (n+3) ⟶ P.complex.X n) |
| variable (hp : ∀ n, p (n+1) ≫ P.complex.d (n+1) n = P.complex.d (n+4) (n+3) ≫ p n) |
|
|
| omit [HasDerivedCategory (ModuleCat.{u} A)] [∀ n, Module.Finite A (P.complex.X n)] in |
| include hp in |
| theorem firstFactor_closed (i : ℕ) (f : P.complex.X i →ₗ[A] U) |
| (hf : f.comp (P.complex.d (i+1) i).hom=0) : |
| (f.comp (p i).hom).comp (P.complex.d (i+4) (i+3)).hom=0 := |
| congrArg ModuleCat.Hom.hom (ResolutionProducts.cubicPrecomp_closed P p hp |
| (ModuleCat.ofHom (X := P.complex.X i) (Y := U) f) (ModuleCat.hom_ext hf)) |
|
|
| omit [Module.Finite k A] [Module.Finite k B] |
| [HasDerivedCategory (ModuleCat.{u} A)] [HasDerivedCategory (ModuleCat.{u} B)] in |
| theorem firstCubicExt_elementary (i j n : ℕ) (h : i+j=n) |
| (f : P.complex.X i →ₗ[A] U) (g : Q.complex.X j →ₗ[B] V) |
| (hf : f.comp (P.complex.d (i+1) i).hom=0) (hg : g.comp (Q.complex.d (j+1) j).hom=0) : |
| (firstCubicExt k A B P Q p hp).comp |
| (elementaryExt k A B P Q U V i j n h f g hf hg) (by omega : 3+n=n+3) = |
| elementaryExt k A B P Q U V (i+3) j (n+3) (by omega) (f.comp (p i).hom) g |
| (firstFactor_closed A P U p hp i f hf) hg := by |
| unfold elementaryExt |
| erw [firstCubicExt_action] |
| have he : firstComparison k A B P Q p n ≫ |
| ModuleCat.ofHom (X := (complex k A B P.complex Q.complex).X n) |
| (Y := moduleObject k A B U V) (elementaryHom k A B P.complex Q.complex U V i j n h f g) = |
| ModuleCat.ofHom (X := (complex k A B P.complex Q.complex).X (n+3)) |
| (Y := moduleObject k A B U V) |
| (elementaryHom k A B P.complex Q.complex U V (i+3) j (n+3) (by omega) (f.comp (p i).hom) g) := |
| ModuleCat.hom_ext (elementary_firstCubic k A B P.complex Q.complex U V p i j n h f g) |
| exact extMk_of_eq (projectiveResolution k A B P Q) _ _ _ _ he |
|
|
| variable (r : ∀ n, Q.complex.X (n+3) ⟶ Q.complex.X n) |
| variable (hr : ∀ n, r (n+1) ≫ Q.complex.d (n+1) n = Q.complex.d (n+4) (n+3) ≫ r n) |
|
|
| omit [HasDerivedCategory (ModuleCat.{u} B)] [∀ n, Module.Finite B (Q.complex.X n)] in |
| include hr in |
| theorem secondFactor_closed (j : ℕ) (g : Q.complex.X j →ₗ[B] V) |
| (hg : g.comp (Q.complex.d (j+1) j).hom=0) : |
| (g.comp (r j).hom).comp (Q.complex.d (j+4) (j+3)).hom=0 := |
| congrArg ModuleCat.Hom.hom (ResolutionProducts.cubicPrecomp_closed Q r hr |
| (ModuleCat.ofHom (X := Q.complex.X j) (Y := V) g) (ModuleCat.hom_ext hg)) |
|
|
| omit [Module.Finite k A] [Module.Finite k B] |
| [HasDerivedCategory (ModuleCat.{u} A)] [HasDerivedCategory (ModuleCat.{u} B)] in |
| theorem secondCubicExt_elementary (i j n : ℕ) (h : i+j=n) |
| (f : P.complex.X i →ₗ[A] U) (g : Q.complex.X j →ₗ[B] V) |
| (hf : f.comp (P.complex.d (i+1) i).hom=0) (hg : g.comp (Q.complex.d (j+1) j).hom=0) : |
| (secondCubicExt k A B P Q r hr).comp |
| (elementaryExt k A B P Q U V i j n h f g hf hg) (by omega : 3+n=n+3) = |
| elementaryExt k A B P Q U V i (j+3) (n+3) (by omega) f (g.comp (r j).hom) |
| hf (secondFactor_closed B Q V r hr j g hg) := by |
| unfold elementaryExt |
| erw [secondCubicExt_action] |
| have he : secondComparison k A B P Q r n ≫ |
| ModuleCat.ofHom (X := (complex k A B P.complex Q.complex).X n) |
| (Y := moduleObject k A B U V) (elementaryHom k A B P.complex Q.complex U V i j n h f g) = |
| ModuleCat.ofHom (X := (complex k A B P.complex Q.complex).X (n+3)) |
| (Y := moduleObject k A B U V) |
| (elementaryHom k A B P.complex Q.complex U V i (j+3) (n+3) (by omega) f (g.comp (r j).hom)) := |
| ModuleCat.hom_ext (elementary_secondCubic k A B P.complex Q.complex U V r i j n h f g) |
| exact extMk_of_eq (projectiveResolution k A B P Q) _ _ _ _ he |
|
|
| end ArExplicit.ExternalTensor |
|
|
| end |
|
|
| end OAI |
|
|
