| import OAI.Algebra.AuslanderReiten.Ext.AExtPolynomial |
| import OAI.Algebra.AuslanderReiten.Ext.ExtKunnethActions |
| import OAI.Algebra.AuslanderReiten.Bar.ModuleBarTensorClass |
| import OAI.Algebra.AuslanderReiten.Bar.ModuleBarCubic |
| import OAI.Algebra.AuslanderReiten.Tensor.ExternalTensorScalarProducts |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open CategoryTheory CategoryTheory.Abelian |
| open scoped TensorProduct ModuleCat.Algebra |
| namespace ArExplicit.Resolution |
| universe u |
| variable {k : Type u} [Field k] [CharP k 2] (q : k) |
| local notation "T" => ArExplicit.T k q |
| local notation "S" => ModuleCat.of T (Simple q) |
| attribute [local instance] moduleCatBaseModule moduleCatBaseTower |
| variable [HasDerivedCategory (ModuleCat.{u} (ArExplicit.T k q))] |
| variable [HasDerivedCategory (ModuleCat.{u} (ArExplicit.T k q ⊗[k] ArExplicit.T k q))] |
|
|
| abbrev aFactorResolution := ModuleBar.nativeResolution k T (Simple q) |
| local instance aFactorFinite (i : ℕ) : Module.Finite T ((aFactorResolution q).complex.X i) := |
| ModuleBar.nativeResolution_finite k T (Simple q) i |
|
|
| abbrev aFactorCubic := ModuleBar.nativeCubicMap k T (Simple q) (CocycleData.p q) |
| abbrev aFactorCubic_comm := ModuleBar.nativeCubicMap_comm k T (Simple q) |
| (CocycleData.p q) (CocycleData.p_isCocycle3 q) |
|
|
| omit [HasDerivedCategory (ModuleCat.{u} (ArExplicit.T k q))] |
| [HasDerivedCategory (ModuleCat.{u} (ArExplicit.T k q ⊗[k] ArExplicit.T k q))] in |
| theorem aFactorExtClass : |
| ModuleBar.extClass3 k T (Simple q) (CocycleData.p q) (CocycleData.p_isCocycle3 q) = |
| tExplicitP q := rfl |
|
|
| abbrev aCanonicalFirst := |
| ModuleBar.nativeTensorFirst k T T (Simple q) (Simple q) |
| (CocycleData.p q) (CocycleData.p_isCocycle3 q) |
|
|
| abbrev aCanonicalSecond := |
| ModuleBar.nativeTensorSecond k T T (Simple q) (Simple q) |
| (CocycleData.p q) (CocycleData.p_isCocycle3 q) |
|
|
| abbrev aFirst := |
| ModuleBar.tensorFirst k T T (Simple q) (Simple q) |
| (CocycleData.p q) (CocycleData.p_isCocycle3 q) |
|
|
| abbrev aSecond := |
| ModuleBar.tensorSecond k T T (Simple q) (Simple q) |
| (CocycleData.p q) (CocycleData.p_isCocycle3 q) |
|
|
| omit [HasDerivedCategory (ModuleCat.{u} (ArExplicit.T k q ⊗[k] ArExplicit.T k q))] in |
| theorem aFactorCubic_class : |
| ResolutionProducts.endExt (aFactorResolution q) |
| (ResolutionProducts.natCubicCocycle (k := k) (aFactorResolution q) |
| (aFactorCubic q) (aFactorCubic_comm q)) = tExplicitP q := |
| (ModuleBar.nativeCubicMap_class k T (Simple q) (CocycleData.p q) |
| (CocycleData.p_isCocycle3 q)).trans (aFactorExtClass q) |
|
|
| theorem aCanonicalFirst_action (n : ℕ) |
| (x : Ext (ExternalTensor.canonicalTensorObject k T T (Simple q) (Simple q)) |
| (ExternalTensor.canonicalTensorObject k T T (Simple q) (Simple q)) n) : |
| aCanonicalKunnethEquiv q (n+3) ((aCanonicalFirst q).comp x (by omega)) = |
| ExternalTensor.coefficientFirst k T T S S S S (tExplicitP q) n |
| (aCanonicalKunnethEquiv q n x) := by |
| have ht := ModuleBar.nativeTensorFirst_action_of_class k T T (Simple q) (Simple q) |
| (CocycleData.p q) (CocycleData.p_isCocycle3 q) (tExplicitP q) (aFactorExtClass q) n x |
| dsimp only [aCanonicalKunnethEquiv, aExtKunnethEquiv, aCanonicalFirst, aCanonicalSecond, aFirst, aSecond, tTensorCoefficients, ArExplicit.A.Simple] at ht ⊢ |
| exact ht |
|
|
| theorem aCanonicalSecond_action (n : ℕ) |
| (x : Ext (ExternalTensor.canonicalTensorObject k T T (Simple q) (Simple q)) |
| (ExternalTensor.canonicalTensorObject k T T (Simple q) (Simple q)) n) : |
| aCanonicalKunnethEquiv q (n+3) ((aCanonicalSecond q).comp x (by omega)) = |
| ExternalTensor.coefficientSecond k T T S S S S (tExplicitP q) n |
| (aCanonicalKunnethEquiv q n x) := by |
| have ht := ModuleBar.nativeTensorSecond_action_of_class k T T (Simple q) (Simple q) |
| (CocycleData.p q) (CocycleData.p_isCocycle3 q) (tExplicitP q) (aFactorExtClass q) n x |
| dsimp only [aCanonicalKunnethEquiv, aExtKunnethEquiv, aCanonicalFirst, aCanonicalSecond, aFirst, aSecond, tTensorCoefficients, ArExplicit.A.Simple] at ht ⊢ |
| exact ht |
|
|
| theorem aFirst_kunneth (n : ℕ) (x : ASelfExt q n) : |
| type_of% (ModuleBar.tensorFirst_action_of_class k T T (Simple q) (Simple q) |
| (CocycleData.p q) (CocycleData.p_isCocycle3 q) (tExplicitP q) (aFactorExtClass q) n x) := |
| ModuleBar.tensorFirst_action_of_class k T T (Simple q) (Simple q) |
| (CocycleData.p q) (CocycleData.p_isCocycle3 q) (tExplicitP q) (aFactorExtClass q) n x |
|
|
| theorem aSecond_kunneth (n : ℕ) (x : ASelfExt q n) : |
| type_of% (ModuleBar.tensorSecond_action_of_class k T T (Simple q) (Simple q) |
| (CocycleData.p q) (CocycleData.p_isCocycle3 q) (tExplicitP q) (aFactorExtClass q) n x) := |
| ModuleBar.tensorSecond_action_of_class k T T (Simple q) (Simple q) |
| (CocycleData.p q) (CocycleData.p_isCocycle3 q) (tExplicitP q) (aFactorExtClass q) n x |
|
|
| end ArExplicit.Resolution |
|
|
| end |
|
|
| end OAI |
|
|
