| import OAI.Algebra.AuslanderReiten.Ext.ExternalComplexIso |
| import OAI.Algebra.AuslanderReiten.Ext.ExternalCubicInclusion |
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|
| namespace OAI |
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| noncomputable section |
| namespace ArExplicit.ExternalTensor |
| open CategoryTheory CategoryTheory.Limits |
| open scoped TensorProduct |
| universe u |
| variable (k A B : Type u) [Field k] [Ring A] [Ring B] [Algebra k A] [Algebra k B] |
| attribute [local instance] moduleCatBaseModule moduleCatBaseTower |
| variable {P P' : ChainComplex (ModuleCat.{u} A) ℕ} |
| {Q Q' : ChainComplex (ModuleCat.{u} B) ℕ} |
| variable (f : P ⟶ P') (g : Q ⟶ Q') |
| variable (p : ∀ n, P.X (n+3) ⟶ P.X n) (p' : ∀ n, P'.X (n+3) ⟶ P'.X n) |
| variable (r : ∀ n, Q.X (n+3) ⟶ Q.X n) (r' : ∀ n, Q'.X (n+3) ⟶ Q'.X n) |
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|
| theorem firstCubic_complexMap |
| (hp : ∀ n, p n ≫ f.f n = f.f (n+3) ≫ p' n) (n : ℕ) : |
| ModuleCat.ofHom (firstCubic k A B P Q p n) ≫ (complexMap k A B f g).f n = |
| (complexMap k A B f g).f (n+3) ≫ ModuleCat.ofHom (firstCubic k A B P' Q' p' n) := by |
| apply hom_ext k A B P Q |
| intro i j hij |
| erw [← Category.assoc, inclusion_firstCubic k A B P Q p i j n hij, |
| ← Category.assoc, inclusion_complexMap k A B f g i j (n+3) hij, |
| Category.assoc, inclusion_firstCubic k A B P' Q' p' i j n hij] |
| by_cases hi : i<3 |
| · have hi : i=0 ∨ i=1 ∨ i=2 := by omega |
| rcases hi with rfl | rfl | rfl <;> change (0 : _ ⟶ _) ≫ _ = _ ≫ (0 : _ ⟶ _) |
| all_goals simp only [zero_comp, comp_zero] |
| · obtain ⟨a,ha⟩ : ∃ a, i=a+3 := ⟨i-3, by omega⟩ |
| subst i |
| erw [firstCubicEntry_high k A B P Q p a j n hij, |
| firstCubicEntry_high k A B P' Q' p' a j n hij, |
| Category.assoc, inclusion_complexMap k A B f g a j n (by omega), |
| ← Category.assoc, ← Category.assoc, |
| ← mapHom_comp k A B (p a) (f.f a) (𝟙 (Q.X j)) (g.f j), |
| ← mapHom_comp k A B (f.f (a+3)) (p' a) (g.f j) (𝟙 (Q'.X j)), |
| Category.id_comp] |
| simpa only [Category.comp_id] using congrArg |
| (fun z => mapHom k A B z (g.f j) ≫ inclusion k A B P' Q' a j n (by omega)) (hp a) |
|
|
| theorem secondCubic_complexMap |
| (hr : ∀ n, r n ≫ g.f n = g.f (n+3) ≫ r' n) (n : ℕ) : |
| ModuleCat.ofHom (secondCubic k A B P Q r n) ≫ (complexMap k A B f g).f n = |
| (complexMap k A B f g).f (n+3) ≫ ModuleCat.ofHom (secondCubic k A B P' Q' r' n) := by |
| apply hom_ext k A B P Q |
| intro i j hij |
| erw [← Category.assoc, inclusion_secondCubic k A B P Q r i j n hij, |
| ← Category.assoc, inclusion_complexMap k A B f g i j (n+3) hij, |
| Category.assoc, inclusion_secondCubic k A B P' Q' r' i j n hij] |
| by_cases hj : j<3 |
| · have hj : j=0 ∨ j=1 ∨ j=2 := by omega |
| rcases hj with rfl | rfl | rfl <;> change (0 : _ ⟶ _) ≫ _ = _ ≫ (0 : _ ⟶ _) |
| all_goals simp only [zero_comp, comp_zero] |
| · obtain ⟨a,ha⟩ : ∃ a, j=a+3 := ⟨j-3, by omega⟩ |
| subst j |
| erw [secondCubicEntry_high k A B P Q r i a n hij, |
| secondCubicEntry_high k A B P' Q' r' i a n hij, |
| Category.assoc, inclusion_complexMap k A B f g i a n (by omega), |
| ← Category.assoc, ← Category.assoc, |
| ← mapHom_comp k A B (𝟙 (P.X i)) (f.f i) (r a) (g.f a), |
| ← mapHom_comp k A B (f.f i) (𝟙 (P'.X i)) (g.f (a+3)) (r' a), |
| Category.id_comp] |
| simpa only [Category.comp_id] using congrArg |
| (fun z => mapHom k A B (f.f i) z ≫ inclusion k A B P' Q' i a n (by omega)) (hr a) |
|
|
| end ArExplicit.ExternalTensor |
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| end |
|
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| end OAI |
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