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d7f9a33 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 | 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
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