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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 46 47 | import OAI.Algebra.AuslanderReiten.Ext.CubicExtAction
namespace OAI
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)
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
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]
end ArExplicit.ResolutionProducts
end
end OAI
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