| import OAI.Algebra.AuslanderReiten.Resolutions.BinaryKoszul |
| import OAI.Algebra.AuslanderReiten.Ext.ExtCentralGeneration |
| |
| namespace OAI |
|
|
| noncomputable section |
| open CategoryTheory CategoryTheory.Abelian |
| open scoped ModuleCat.Algebra |
| namespace ArExplicit.ResolutionProducts |
| universe u |
| variable {k R : Type u} [Field k] [Ring R] [Algebra k R] |
| variable {X : ModuleCat.{u} R} |
| variable (a b : Ext X X 3) |
| variable (e : ∀ n : ℕ, Ext X X (3*n) ≃ₗ[k] BinaryKoszul.Piece k 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)) |
| (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 ha hb |
|
|
| theorem cubicCommute_of_coefficients : 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 (e 2).injective |
| rw [ha, hb, h0a, h0b] |
| exact LinearMap.congr_fun (BinaryKoszul.mul_commute k 0) _ |
|
|
| theorem cubicGeneration_of_coefficients (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) |
| refine ⟨(e n).symm u, (e n).symm v, ?_⟩ |
| apply (e (n+1)).injective |
| rw [map_add, ha, hb, LinearEquiv.apply_symm_apply, LinearEquiv.apply_symm_apply] |
| exact huv.symm |
|
|
| theorem cubicCentral_of_coefficients |
| (hzero : ∀ x : Ext X X 0, |
| a.comp x (by omega : 3+0=3) = x.comp a (by omega : 0+3=3) ∧ |
| b.comp x (by omega : 3+0=3) = x.comp b (by omega : 0+3=3)) : |
| ∀ (n : ℕ) (x : Ext X X (3*n)), |
| a.comp x (by omega : 3+3*n=3*(n+1)) = x.comp a (by omega) ∧ |
| b.comp x (by omega : 3+3*n=3*(n+1)) = x.comp b (by omega) := |
| cubicCentral_of_generation a b (cubicCommute_of_coefficients a b e ha hb) |
| hzero (cubicGeneration_of_coefficients a b e ha hb) |
|
|
| omit hb in |
| theorem cubicFirst_right_of_coefficients (n : ℕ) (x : Ext X X (3*n)) |
| (hc : a.comp x (by omega : 3+3*n=3*(n+1)) = x.comp a (by omega)) : |
| e (n+1) (x.comp a (by omega)) = BinaryKoszul.mulX k n (e n x) := |
| (congrArg (e (n+1)) hc.symm).trans (ha n x) |
|
|
| omit ha in |
| theorem cubicSecond_right_of_coefficients (n : ℕ) (x : Ext X X (3*n)) |
| (hc : b.comp x (by omega : 3+3*n=3*(n+1)) = x.comp b (by omega)) : |
| e (n+1) (x.comp b (by omega)) = BinaryKoszul.mulY k n (e n x) := |
| (congrArg (e (n+1)) hc.symm).trans (hb n x) |
|
|
| end ArExplicit.ResolutionProducts |
|
|
| end |
|
|
| end OAI |
|
|
