| import OAI.Algebra.AuslanderReiten.HomComplex.FullHomFunctor | |
| namespace OAI | |
| noncomputable section | |
| namespace ArExplicit | |
| open CategoryTheory CochainComplex.HomComplex | |
| variable {C D : Type*} [Category* C] [Category* D] [Preadditive C] [Preadditive D] | |
| variable (Φ : C ⥤ D) [Φ.Additive] | |
| variable (K L : CochainComplex C ℤ) (n : ℤ) | |
| def mapFullHom : CohomologyClass K L n →+ | |
| CohomologyClass ((Φ.mapHomologicalComplex _).obj K) | |
| ((Φ.mapHomologicalComplex _).obj L) n := | |
| CohomologyClass.descAddMonoidHom | |
| ((CohomologyClass.mkAddMonoidHom _ _ _).comp (fullHomCocycleMap Φ K L n)) (by | |
| intro x hx | |
| apply (CohomologyClass.mk_eq_zero_iff _).mpr | |
| obtain ⟨m, hm, y, hy⟩ := hx | |
| refine ⟨m, hm, y.map Φ, ?_⟩ | |
| rw [δ_map, hy] | |
| rfl) | |
| @[simp] theorem mapFullHom_mk (x : Cocycle K L n) : | |
| mapFullHom Φ K L n (CohomologyClass.mk x) = | |
| CohomologyClass.mk (fullHomCocycleMap Φ K L n x) := rfl | |
| theorem mapFullHom_eq [Φ.Full] [Φ.Faithful] : | |
| mapFullHom Φ K L n = fullHomCohomologyMap Φ K L n := rfl | |
| variable {K L n} | |
| theorem mapFullHom_product {M : CochainComplex C ℤ} {m : ℤ} | |
| (x : CohomologyClass K L n) (y : CohomologyClass L M m) : | |
| mapFullHom Φ K M (n+m) (CompleteComparison.fullHomProduct x y) = | |
| CompleteComparison.fullHomProduct (mapFullHom Φ K L n x) | |
| (mapFullHom Φ L M m y) := by | |
| obtain ⟨x, rfl⟩ := CohomologyClass.mk_surjective x | |
| obtain ⟨y, rfl⟩ := CohomologyClass.mk_surjective y | |
| simp only [CompleteComparison.fullHomProduct_mk, mapFullHom_mk] | |
| congr 1 | |
| apply Cocycle.ext | |
| apply Cochain.map_comp | |
| rfl | |
| end ArExplicit | |
| end | |
| end OAI | |