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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 32 33 34 35 36 37 38 39 40 41 42 43 44 45 | 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
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