| import OAI.Algebra.AuslanderReiten.Ext.ExtPolynomialTransfer |
| |
| namespace OAI |
|
|
| noncomputable section |
| open CategoryTheory CategoryTheory.Abelian |
| namespace ArExplicit.ResolutionProducts |
| universe u |
| variable {k R : Type u} [Field k] [Ring R] |
| variable {X : ModuleCat.{u} R} |
| variable (a b : Ext X X 3) |
| variable (e : ∀ n : ℕ, Ext X X (3*n) → BinaryKoszul.Piece k n) |
| variable (hi : ∀ n, Function.Injective (e n)) |
| variable (ha : ∀ (n : ℕ) (x : Ext X X (3*n)), |
| e (n+1) (a.comp x (by omega)) = BinaryKoszul.mulX k n (e n x)) |
| variable (hb : ∀ (n : ℕ) (x : Ext X X (3*n)), |
| e (n+1) (b.comp x (by omega)) = BinaryKoszul.mulY k n (e n x)) |
| include hi ha hb |
|
|
| theorem cubicCommute_of_functions : a.comp b rfl = b.comp a rfl := by |
| have h0a := ha 0 (Ext.mk₀ (𝟙 X)) |
| have h0b := hb 0 (Ext.mk₀ (𝟙 X)) |
| simp only [Ext.comp_mk₀_id] at h0a h0b |
| apply hi 2 |
| rw [ha, hb, h0a, h0b] |
| exact LinearMap.congr_fun (BinaryKoszul.mul_commute k 0) _ |
|
|
| theorem cubicGeneration_of_functions |
| (hs : ∀ n, Function.Surjective (e n)) |
| (hadd : ∀ n x y, e n (x+y) = e n x + e n y) |
| (n : ℕ) (x : Ext X X (3*(n+1))) : |
| ∃ u v : Ext X X (3*n), x = a.comp u (by omega) + b.comp v (by omega) := by |
| obtain ⟨⟨u,v⟩,huv⟩ := BinaryKoszul.row_surjective k n (e (n+1) x) |
| obtain ⟨u',hu⟩ := hs n u |
| obtain ⟨v',hv⟩ := hs n v |
| refine ⟨u',v', hi (n+1) ?_⟩ |
| rw [hadd,ha,hb,hu,hv] |
| exact huv.symm |
|
|
| end ArExplicit.ResolutionProducts |
|
|
| end |
|
|
| end OAI |
|
|
