| import OAI.Algebra.AuslanderReiten.HomComplex.FullHomZeroRepresentative |
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| namespace OAI |
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| noncomputable section |
| open CategoryTheory CochainComplex.HomComplex |
| namespace ArExplicit |
| universe u v w |
| variable {C : Type w} [Category.{v} C] [Preadditive C] |
| variable {K L M : CochainComplex C ℤ} |
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|
| theorem fullHomPostcomp_ofHom (f : K ⟶ L) (g : L ⟶ M) : |
| fullHomPostcomp g 0 (CohomologyClass.mk (Cocycle.ofHom f)) = |
| CohomologyClass.mk (Cocycle.ofHom (f ≫ g)) := by |
| rw [← fullHomPostcomp_identity f, fullHomPostcomp_comp, fullHomPostcomp_identity] |
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| variable (k : Type u) [CommRing k] [Linear k C] |
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| include k in |
| theorem fullHomZero_projection_inverse (f : K ⟶ L) (g : L ⟶ K) |
| (e : (HomotopyCategory.quotient C _).obj L ≅ (HomotopyCategory.quotient C _).obj K) |
| (he : (HomotopyCategory.quotient C _).map g = e.hom) |
| (h : fullHomPostcomp g 0 (CohomologyClass.mk (Cocycle.ofHom f)) = |
| CohomologyClass.mk (Cocycle.ofHom (𝟙 K))) : |
| (HomotopyCategory.quotient C _).map f = e.inv := by |
| rw [fullHomPostcomp_ofHom] at h |
| have hh := (fullHomZero_class_eq_iff k (f ≫ g) (𝟙 K)).mp h |
| rw [CategoryTheory.Functor.map_comp, he, CategoryTheory.Functor.map_id] at hh |
| apply (cancel_mono e.hom).1 |
| exact hh.trans e.inv_hom_id.symm |
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|
| end ArExplicit |
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| end |
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| end OAI |
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|
