| import OAI.Algebra.AuslanderReiten.Tensor.ExternalTensorDifferential |
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| namespace OAI |
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| noncomputable section |
| namespace ArExplicit.ExternalTensor |
| open CategoryTheory HomologicalComplex |
| 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' P'' : ChainComplex (ModuleCat.{u} A) ℕ} |
| {Q Q' Q'' : ChainComplex (ModuleCat.{u} B) ℕ} |
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| abbrev complexMap (f : P ⟶ P') (g : Q ⟶ Q') := |
| mapBifunctorMap f g (bifunctor k A B) (ComplexShape.down ℕ) |
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| @[reassoc] theorem inclusion_complexMap (f : P ⟶ P') (g : Q ⟶ Q') |
| (i j n : ℕ) (h : i+j=n) : |
| inclusion k A B P Q i j n h ≫ (complexMap k A B f g).f n = |
| mapHom k A B (f.f i) (g.f j) ≫ inclusion k A B P' Q' i j n h := by |
| change ιMapBifunctor P Q (bifunctor k A B) (ComplexShape.down ℕ) i j n h ≫ |
| (mapBifunctorMap f g (bifunctor k A B) (ComplexShape.down ℕ)).f n = _ |
| erw [ι_mapBifunctorMap] |
| change mapHom k A B (f.f i) (𝟙 (Q.X j)) ≫ |
| mapHom k A B (𝟙 (P'.X i)) (g.f j) ≫ _ = _ |
| erw [← Category.assoc, ← mapHom_comp, Category.comp_id, Category.id_comp] |
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| @[simp] theorem complexMap_id : complexMap k A B (𝟙 P) (𝟙 Q) = 𝟙 (complex k A B P Q) := by |
| apply HomologicalComplex.Hom.ext |
| funext n |
| apply hom_ext k A B P Q |
| intro i j h |
| erw [inclusion_complexMap] |
| simp only [HomologicalComplex.id_f, mapHom_id, Category.id_comp, Category.comp_id] |
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| theorem complexMap_comp (f : P ⟶ P') (f' : P' ⟶ P'') (g : Q ⟶ Q') (g' : Q' ⟶ Q'') : |
| complexMap k A B (f ≫ f') (g ≫ g') = complexMap k A B f g ≫ complexMap k A B f' g' := by |
| apply HomologicalComplex.Hom.ext |
| funext n |
| apply hom_ext k A B P Q |
| intro i j h |
| erw [inclusion_complexMap k A B (f ≫ f') (g ≫ g') i j n h] |
| change mapHom k A B ((f ≫ f').f i) ((g ≫ g').f j) ≫ _ = |
| inclusion k A B P Q i j n h ≫ ((complexMap k A B f g).f n ≫ (complexMap k A B f' g').f n) |
| erw [← Category.assoc, inclusion_complexMap k A B f g i j n h, Category.assoc, |
| inclusion_complexMap k A B f' g' i j n h, HomologicalComplex.comp_f, |
| HomologicalComplex.comp_f, mapHom_comp k A B (f.f i) (f'.f i) (g.f j) (g'.f j), Category.assoc] |
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| def complexIso (e : P ≅ P') (f : Q ≅ Q') : complex k A B P Q ≅ complex k A B P' Q' where |
| hom := complexMap k A B e.hom f.hom |
| inv := complexMap k A B e.inv f.inv |
| hom_inv_id := by rw [← complexMap_comp, e.hom_inv_id, f.hom_inv_id, complexMap_id] |
| inv_hom_id := by rw [← complexMap_comp, e.inv_hom_id, f.inv_hom_id, complexMap_id] |
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| end ArExplicit.ExternalTensor |
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| end |
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| end OAI |
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