| import Mathlib.Algebra.Homology.DerivedCategory.Ext.Linear |
| import Mathlib.Algebra.Category.ModuleCat.Ext.HasExt |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open CategoryTheory CategoryTheory.Abelian |
| namespace ArExplicit.ResolutionProducts |
| universe u |
| variable {R : Type u} [Ring R] {X : ModuleCat.{u} R} |
| variable {D₀ D₁ : Type u} [AddCommGroup D₀] [AddCommGroup D₁] |
|
|
| theorem generationOfAddEquiv {n : ℕ} (a b : Ext X X 3) |
| (e₀ : Ext X X n ≃+ D₀) (e₁ : Ext X X (n+3) ≃+ D₁) |
| (f g : D₀ → D₁) |
| (ha : ∀ x, e₁ (a.comp x (by omega)) = f (e₀ x)) |
| (hb : ∀ x, e₁ (b.comp x (by omega)) = g (e₀ x)) |
| (hgen : ∀ z : D₁, ∃ u v, z = f u + g v) |
| (x : Ext X X (n+3)) : |
| ∃ u v : Ext X X n, x = a.comp u (by omega) + b.comp v (by omega) := by |
| obtain ⟨u,v,huv⟩ := hgen (e₁ x) |
| refine ⟨e₀.symm u,e₀.symm v,e₁.injective ?_⟩ |
| rw [map_add,ha,hb,AddEquiv.apply_symm_apply,AddEquiv.apply_symm_apply] |
| exact huv |
|
|
| end ArExplicit.ResolutionProducts |
|
|
| end |
|
|
| end OAI |
|
|
