| import OAI.Algebra.AuslanderReiten.Resolutions.ResolutionExtAction |
| import OAI.Algebra.AuslanderReiten.Ext.NatCubicExt |
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| namespace OAI |
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| noncomputable section |
| open CategoryTheory CategoryTheory.Abelian |
| open CochainComplex CochainComplex.HomComplex |
| 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 Y : ModuleCat.{u} R} (P : ProjectiveResolution X) |
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| def closedCochain {n : ℕ} (f : P.complex.X n ⟶ Y) |
| (hf : P.complex.d (n+1) n ≫ f=0) : |
| Cocycle P.cochainComplex ((CochainComplex.singleFunctor (ModuleCat R) 0).obj Y) (n : ℤ) := |
| Cocycle.toSingleMk ((P.cochainComplexXIso (-(n : ℤ)) n rfl).hom ≫ f) (by omega) |
| (-((n+1 : ℕ) : ℤ)) (by omega) (by |
| rw [P.cochainComplex_d _ _ (n+1) n rfl rfl] |
| simp only [Category.assoc, Iso.inv_hom_id_assoc, hf, CategoryTheory.Limits.comp_zero]) |
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| omit [HasDerivedCategory (ModuleCat.{u} R)] in |
| @[simp] theorem extOf_closedCochain {n : ℕ} (f : P.complex.X n ⟶ Y) |
| (hf : P.complex.d (n+1) n ≫ f=0) : |
| extOfCocycle P (closedCochain P f hf) = P.extMk f (n+1) rfl hf := rfl |
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| variable (p : ∀ n, P.complex.X (n+3) ⟶ P.complex.X n) |
| variable (hp : ∀ n, p (n+1) ≫ P.complex.d (n+1) n = P.complex.d (n+4) (n+3) ≫ p n) |
| include hp |
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| omit [HasDerivedCategory (ModuleCat.{u} R)] in |
| theorem cubicPrecomp_closed {n : ℕ} (f : P.complex.X n ⟶ Y) |
| (hf : P.complex.d (n+1) n ≫ f=0) : |
| P.complex.d (n+4) (n+3) ≫ (p n ≫ f)=0 := by |
| rw [← Category.assoc, ← hp n, Category.assoc, hf, CategoryTheory.Limits.comp_zero] |
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|
| include k in |
| omit [HasDerivedCategory (ModuleCat.{u} R)] in |
| theorem cubicPrecomp_cochain {n : ℕ} (f : P.complex.X n ⟶ Y) |
| (hf : P.complex.d (n+1) n ≫ f=0) : |
| (closedCochain P (p n ≫ f) (cubicPrecomp_closed P p hp f hf)).val = |
| (natCubicCocycle (k := k) P p hp).val.comp (closedCochain P f hf).val |
| (by omega : (3 : ℤ)+(n : ℤ)=((n+3 : ℕ) : ℤ)) := by |
| apply (Cochain.toSingleEquiv (show -((n+3 : ℕ) : ℤ)+((n+3 : ℕ) : ℤ)=0 by omega)).injective |
| change (Cochain.toSingleMk ((P.cochainComplexXIso (-((n+3 : ℕ) : ℤ)) (n+3) rfl).hom ≫ p n ≫ f) |
| (by omega)).v (-((n+3 : ℕ) : ℤ)) 0 (by omega) ≫ |
| (HomologicalComplex.singleObjXSelf (.up ℤ) 0 Y).hom = |
| (((natCubicCocycle (k := k) P p hp).val.comp (closedCochain P f hf).val (by omega)).v |
| (-((n+3 : ℕ) : ℤ)) 0 (by omega)) ≫ |
| (HomologicalComplex.singleObjXSelf (.up ℤ) 0 Y).hom |
| rw [Cochain.toSingleMk_v] |
| erw [Cochain.comp_v _ _ _ _ (-(n : ℤ)) 0 (by omega) (by omega)] |
| change _ = ((natCubicCochain P p).v (-((n+3 : ℕ) : ℤ)) (-(n : ℤ)) (by omega) ≫ |
| (Cochain.toSingleMk ((P.cochainComplexXIso (-(n : ℤ)) n rfl).hom ≫ f) |
| (by omega)).v (-(n : ℤ)) 0 (by omega)) ≫ _ |
| rw [Cochain.toSingleMk_v] |
| have h := natCubicCochain_v P p n |
| rw [h] |
| let e := P.cochainComplexXIso (-(n : ℤ)) n rfl |
| let e₃ := P.cochainComplexXIso (-((n+3 : ℕ) : ℤ)) (n+3) rfl |
| let s := HomologicalComplex.singleObjXSelf (.up ℤ) 0 Y |
| exact (congrArg (fun z => ((e₃.hom ≫ p n ≫ z ≫ f) ≫ s.inv) ≫ s.hom) |
| e.inv_hom_id).symm |
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| include k in |
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| theorem cubic_endExt_comp_extMk {n : ℕ} (f : P.complex.X n ⟶ Y) |
| (hf : P.complex.d (n+1) n ≫ f=0) : |
| (endExt P (natCubicCocycle (k := k) P p hp)).comp (P.extMk f (n+1) rfl hf) |
| (by omega : 3+n=n+3) = |
| P.extMk (p n ≫ f) (n+4) rfl (cubicPrecomp_closed P p hp f hf) := by |
| exact endExt_comp_extOfCocycle P (n := 3) (m := n) (r := n+3) (natCubicCocycle (k := k) P p hp) |
| (closedCochain P f hf) |
| (closedCochain P (p n ≫ f) (cubicPrecomp_closed P p hp f hf)) (by omega) |
| (cubicPrecomp_cochain (k := k) P p hp f hf) |
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| end ArExplicit.ResolutionProducts |
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| end |
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| end OAI |
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