| import OAI.Algebra.AuslanderReiten.Ext.AExtActions |
| import OAI.Algebra.AuslanderReiten.Ext.TTensorActions |
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| namespace OAI |
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| noncomputable section |
| open CategoryTheory CategoryTheory.Abelian |
| open scoped TensorProduct ModuleCat.Algebra |
| namespace ArExplicit.ModuleBar |
| universe u v |
| variable (k R S M N : Type u) [Field k] [CharP k 2] |
| [Ring R] [Algebra k R] [Module.Finite k R] [Ring S] [Algebra k S] [Module.Finite k S] |
| [AddCommGroup M] [Module R M] [Module k M] [IsScalarTower k R M] [Module.Finite k M] |
| [AddCommGroup N] [Module S N] [Module k N] [IsScalarTower k S N] [Module.Finite k N] |
| [HasDerivedCategory (ModuleCat.{u} R)] |
| [HasDerivedCategory (ModuleCat.{u} (R ⊗[k] S))] |
| variable (p : AbsoluteBar.Cochain3 k R) (hp : AbsoluteBar.IsCocycle3 k R p) |
| private theorem tensorFirst_action_map (a : Ext (ModuleCat.of R M) (ModuleCat.of R M) 3) |
| (ha : extClass3 k R M p hp = a) (n : ℕ) {W : Sort v} |
| (C : ExternalTensor.extCoefficients k R S (ModuleCat.of R M) (ModuleCat.of R M) |
| (ModuleCat.of S N) (ModuleCat.of S N) (n+3) → W) |
| (D : ExternalTensor.extCoefficients k R S (ModuleCat.of R M) (ModuleCat.of R M) |
| (ModuleCat.of S N) (ModuleCat.of S N) n → W) |
| (hC : ∀ z, C (ExternalTensor.coefficientFirst k R S (ModuleCat.of R M) (ModuleCat.of R M) |
| (ModuleCat.of S N) (ModuleCat.of S N) a n z) = D z) |
| (x : Ext (ExternalTensor.moduleObject k R S M N) (ExternalTensor.moduleObject k R S M N) n) : |
| C (tensorKunneth k R S M N (n+3) ((tensorFirst k R S M N p hp).comp x (by omega))) = |
| D (tensorKunneth k R S M N n x) := |
| (congrArg C (tensorFirst_action_of_class k R S M N p hp a ha n x)).trans (hC _) |
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| variable [HasDerivedCategory (ModuleCat.{u} S)] |
| variable (r : AbsoluteBar.Cochain3 k S) (hr : AbsoluteBar.IsCocycle3 k S r) |
| omit [HasDerivedCategory (ModuleCat.{u} R)] in |
| private theorem tensorSecond_action_map (a : Ext (ModuleCat.of S N) (ModuleCat.of S N) 3) |
| (ha : extClass3 k S N r hr = a) (n : ℕ) {W : Sort v} |
| (C : ExternalTensor.extCoefficients k R S (ModuleCat.of R M) (ModuleCat.of R M) |
| (ModuleCat.of S N) (ModuleCat.of S N) (n+3) → W) |
| (D : ExternalTensor.extCoefficients k R S (ModuleCat.of R M) (ModuleCat.of R M) |
| (ModuleCat.of S N) (ModuleCat.of S N) n → W) |
| (hC : ∀ z, C (ExternalTensor.coefficientSecond k R S (ModuleCat.of R M) (ModuleCat.of R M) |
| (ModuleCat.of S N) (ModuleCat.of S N) a n z) = D z) |
| (x : Ext (ExternalTensor.moduleObject k R S M N) (ExternalTensor.moduleObject k R S M N) n) : |
| C (tensorKunneth k R S M N (n+3) ((tensorSecond k R S M N r hr).comp x (by omega))) = |
| D (tensorKunneth k R S M N n x) := |
| (congrArg C (tensorSecond_action_of_class k R S M N r hr a ha n x)).trans (hC _) |
| end ArExplicit.ModuleBar |
| namespace ArExplicit.Resolution |
| universe u |
| variable {k : Type u} [Field k] [CharP k 2] (q : k) |
| variable [HasDerivedCategory (ModuleCat.{u} (ArExplicit.T k q))] |
| variable [HasDerivedCategory (ModuleCat.{u} (ArExplicit.T k q ⊗[k] ArExplicit.T k q))] |
| variable (hq : q ≠ 0) (hq1 : 1+q ≠ 0) (h : NoPositivePowerOne q) |
| include hq hq1 h |
| local notation "T" => ArExplicit.T k q |
| private abbrev firstCoefficientProof (n : ℕ) (x : ASelfExt q (3*n)) := |
| ModuleBar.tensorFirst_action_map k T T (Simple q) (Simple q) |
| (CocycleData.p q) (CocycleData.p_isCocycle3 q) (tExplicitP q) (aFactorExtClass q) (3*n) |
| (tTensorCoefficientEquiv q hq hq1 h (n+1)) |
| (fun z => BinaryKoszul.mulX k n (tTensorCoefficientEquiv q hq hq1 h n z)) |
| (tTensorCoefficient_first q hq hq1 h n) x |
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| theorem aExtCoefficient_first (n : ℕ) (x : ASelfExt q (3*n)) : |
| type_of% (firstCoefficientProof q hq hq1 h n x) := |
| firstCoefficientProof q hq hq1 h n x |
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| private abbrev secondCoefficientProof (n : ℕ) (x : ASelfExt q (3*n)) := |
| ModuleBar.tensorSecond_action_map k T T (Simple q) (Simple q) |
| (CocycleData.p q) (CocycleData.p_isCocycle3 q) (tExplicitP q) (aFactorExtClass q) (3*n) |
| (tTensorCoefficientEquiv q hq hq1 h (n+1)) |
| (fun z => BinaryKoszul.mulY k n (tTensorCoefficientEquiv q hq hq1 h n z)) |
| (tTensorCoefficient_second q hq hq1 h n) x |
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| theorem aExtCoefficient_second (n : ℕ) (x : ASelfExt q (3*n)) : |
| type_of% (secondCoefficientProof q hq hq1 h n x) := |
| secondCoefficientProof q hq hq1 h n x |
| end ArExplicit.Resolution |
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| end |
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| end OAI |
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