| import OAI.Algebra.AuslanderReiten.Ext.CubicExtAction |
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| namespace OAI |
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| noncomputable section |
| open CategoryTheory CategoryTheory.Abelian |
| namespace ArExplicit.ResolutionProducts |
| universe u |
| variable {k R : Type u} [Field k] [Ring R] [Algebra k R] [CharP k 2] |
| variable [HasDerivedCategory (ModuleCat.{u} R)] |
| variable {X : ModuleCat.{u} R} (P : ProjectiveResolution X) |
| variable (f g : ∀ n, P.complex.X (n+3) ⟶ P.complex.X n) |
| variable (hf : ∀ n, f (n+1) ≫ P.complex.d (n+1) n = P.complex.d (n+4) (n+3) ≫ f n) |
| variable (hg : ∀ n, g (n+1) ≫ P.complex.d (n+1) n = P.complex.d (n+4) (n+3) ≫ g n) |
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| omit [HasDerivedCategory (ModuleCat.{u} R)] in |
| private theorem extMk_eq_of_map_eq {n : ℕ} (a b : P.complex.X n ⟶ X) |
| (ha : P.complex.d (n+1) n ≫ a = 0) (hb : P.complex.d (n+1) n ≫ b = 0) |
| (hab : a = b) : P.extMk a (n+1) rfl ha = P.extMk b (n+1) rfl hb := by |
| subst b |
| rfl |
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| theorem cubic_endExt_commute (hfg : f 3 ≫ g 0 = g 3 ≫ f 0) : |
| (endExt P (natCubicCocycle (k := k) P f hf)).comp |
| (endExt P (natCubicCocycle (k := k) P g hg)) rfl = |
| (endExt P (natCubicCocycle (k := k) P g hg)).comp |
| (endExt P (natCubicCocycle (k := k) P f hf)) rfl := by |
| have h₁ := (congrArg (fun y => |
| (endExt P (natCubicCocycle (k := k) P f hf)).comp y (show 3+3=6 from rfl)) |
| (natCubic_endExt_eq (k := k) P g hg)).trans |
| (cubic_endExt_comp_extMk (k := k) P f hf (g 0 ≫ P.π.f 0) |
| (natCubic_augmented_cycle P g hg)) |
| have h₂ := (congrArg (fun y => |
| (endExt P (natCubicCocycle (k := k) P g hg)).comp y (show 3+3=6 from rfl)) |
| (natCubic_endExt_eq (k := k) P f hf)).trans |
| (cubic_endExt_comp_extMk (k := k) P g hg (f 0 ≫ P.π.f 0) |
| (natCubic_augmented_cycle P f hf)) |
| refine h₁.trans ((extMk_eq_of_map_eq P _ _ _ _ ?_).trans h₂.symm) |
| change f 3 ≫ (g 0 ≫ P.π.f 0) = g 3 ≫ (f 0 ≫ P.π.f 0) |
| rw [← Category.assoc, hfg, Category.assoc] |
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| end ArExplicit.ResolutionProducts |
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| end |
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| end OAI |
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