| import OAI.Algebra.AuslanderReiten.HomComplex.FullHomMap |
| import OAI.Algebra.AuslanderReiten.HomComplex.FullHomProductLinear |
| import OAI.Algebra.AuslanderReiten.HomComplex.FullHomComposition |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open CategoryTheory |
| open CochainComplex CochainComplex.HomComplex |
| namespace ArExplicit |
| variable {k : Type*} [CommRing k] |
| variable {C D : Type*} [Category* C] [Category* D] [Preadditive C] [Preadditive D] |
| [Linear k C] [Linear k D] |
| variable (Φ : C ⥤ D) [Φ.Additive] [Φ.Linear k] |
| variable (K L : CochainComplex C ℤ) (n : ℤ) |
|
|
| theorem mapFullHom_smul (a : k) (x : CohomologyClass K L n) : |
| mapFullHom Φ K L n (a • x) = a • mapFullHom Φ K L n x := by |
| obtain ⟨x,rfl⟩ := CohomologyClass.mk_surjective x |
| rw [← CompleteComparison.fullHom_mk_smul, mapFullHom_mk, mapFullHom_mk, |
| ← CompleteComparison.fullHom_mk_smul] |
| congr 1 |
| apply Subtype.ext |
| apply Cochain.ext |
| intro p q hpq |
| exact Φ.map_smul a (x.val.v p q hpq) |
|
|
| def mapFullHomLinear : CohomologyClass K L n →ₗ[k] |
| CohomologyClass ((Φ.mapHomologicalComplex _).obj K) |
| ((Φ.mapHomologicalComplex _).obj L) n where |
| __ := mapFullHom Φ K L n |
| map_smul' := mapFullHom_smul Φ K L n |
|
|
| theorem mapFullHom_identity : |
| mapFullHom Φ K K 0 (CohomologyClass.mk (Cocycle.ofHom (𝟙 K))) = |
| CohomologyClass.mk (Cocycle.ofHom (𝟙 ((Φ.mapHomologicalComplex _).obj K))) := by |
| rw [mapFullHom_mk] |
| congr 1 |
| apply Subtype.ext |
| change (Cochain.ofHom (𝟙 K)).map Φ = |
| Cochain.ofHom (𝟙 ((Φ.mapHomologicalComplex _).obj K)) |
| rw [Cochain.map_ofHom, CategoryTheory.Functor.map_id] |
|
|
| variable {K L n} |
| theorem fullHomPostcomp_smul (f : K ⟶ L) (n : ℤ) (a : k) |
| (x : CohomologyClass K K n) : |
| fullHomPostcomp f n (a • x) = a • fullHomPostcomp f n x := by |
| obtain ⟨x,rfl⟩ := CohomologyClass.mk_surjective x |
| rw [← CompleteComparison.fullHom_mk_smul, fullHomPostcomp_mk, fullHomPostcomp_mk, |
| ← CompleteComparison.fullHom_mk_smul] |
| congr 1 |
| apply Subtype.ext |
| exact Cochain.smul_comp a x.val (Cochain.ofHom f) (add_zero n) |
|
|
| theorem fullHomPrecomp_smul (f : K ⟶ L) (n : ℤ) (a : k) |
| (x : CohomologyClass L L n) : |
| fullHomPrecomp f n (a • x) = a • fullHomPrecomp f n x := by |
| obtain ⟨x,rfl⟩ := CohomologyClass.mk_surjective x |
| rw [← CompleteComparison.fullHom_mk_smul, fullHomPrecomp_mk, fullHomPrecomp_mk, |
| ← CompleteComparison.fullHom_mk_smul] |
| congr 1 |
| apply Subtype.ext |
| exact Cochain.comp_smul (Cochain.ofHom f) a x.val (zero_add n) |
|
|
| theorem fullHomPostcomp_identity (f : K ⟶ L) : |
| fullHomPostcomp f 0 (CohomologyClass.mk (Cocycle.ofHom (𝟙 K))) = |
| CohomologyClass.mk (Cocycle.ofHom f) := by |
| rw [fullHomPostcomp_mk] |
| congr 1 |
| apply Subtype.ext |
| exact Cochain.id_comp _ |
|
|
| theorem fullHomPrecomp_identity (f : K ⟶ L) : |
| fullHomPrecomp f 0 (CohomologyClass.mk (Cocycle.ofHom (𝟙 L))) = |
| CohomologyClass.mk (Cocycle.ofHom f) := by |
| rw [fullHomPrecomp_mk] |
| congr 1 |
| apply Subtype.ext |
| exact Cochain.comp_id _ |
|
|
| end ArExplicit |
|
|
| end |
|
|
| end OAI |
|
|
