| import Mathlib | |
| namespace OAI | |
| section | |
| open CategoryTheory Limits HomologicalComplex | |
| namespace ChainComplex | |
| open scoped _root_.ChainComplex | |
| variable {R:Type} [CommRing R] (K:ChainComplex (ModuleCat.{0} R) ℕ) | |
| variable (n:ℕ) (hn:K.d (n+1) n=0) | |
| noncomputable def positiveπ : K.X (n+1) ⟶ K.homology (n+1) := | |
| K.pOpcycles (n+1) ≫ (K.isoHomologyι (n+1) n (by simp) hn).inv | |
| instance : Epi (positiveπ K n hn) := by dsimp only [positiveπ]; infer_instance | |
| @[reassoc (attr:=simp)] lemma d_positiveπ : K.d (n+2) (n+1) ≫ positiveπ K n hn=0 := by | |
| simp [positiveπ] | |
| noncomputable def positiveDesc {V:ModuleCat.{0} R} (f:K.X (n+1) ⟶ V) | |
| (hf:K.d (n+2) (n+1) ≫ f=0) : K.homology (n+1) ⟶ V := | |
| (K.isoHomologyι (n+1) n (by simp) hn).hom ≫ | |
| K.descOpcycles f (n+2) (by simp) hf | |
| @[reassoc (attr:=simp)] lemma positiveπ_desc {V:ModuleCat.{0} R} (f:K.X (n+1) ⟶ V) | |
| (hf:K.d (n+2) (n+1) ≫ f=0) : positiveπ K n hn ≫ positiveDesc K n hn f hf=f := by | |
| simp [positiveπ,positiveDesc] | |
| end ChainComplex | |
| end | |
| end OAI | |