| import OAI.Algebra.AuslanderReiten.Ext.ExtKunneth | |
| import OAI.Algebra.AuslanderReiten.Ext.TTensorPolynomial | |
| import OAI.Algebra.AuslanderReiten.Tensor.ExternalTensorScalarExt | |
| import OAI.Algebra.AuslanderReiten.Algebras.TensorSimple | |
| import OAI.Algebra.AuslanderReiten.Bar.ModuleBarTensorExt | |
| 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 | |
| abbrev ASelfExt (n : ℕ) := Ext (ArExplicit.A.Simple q) (ArExplicit.A.Simple q) n | |
| abbrev aCanonicalKunnethEquiv (n : ℕ) := | |
| ModuleBar.nativeTensorKunneth k T T (Simple q) (Simple q) n | |
| abbrev aExtKunnethEquiv (n : ℕ) := | |
| ModuleBar.tensorKunneth k T T (Simple q) (Simple q) n | |
| variable [HasDerivedCategory (ModuleCat.{u} (ArExplicit.T k q))] | |
| variable (hq : q ≠ 0) (hq1 : 1+q ≠ 0) (h : NoPositivePowerOne q) | |
| abbrev aExtCoefficientEquiv (n : ℕ) := | |
| (aExtKunnethEquiv q (3*n)).trans (tTensorCoefficientEquiv q hq hq1 h n) | |
| include hq hq1 h in | |
| theorem aSelfExt_zero_of_not_dvd (n : ℕ) (hn : ¬ 3 ∣ n) (x : ASelfExt q n) : x=0 := by | |
| exact (aExtKunnethEquiv q n).map_eq_zero_iff.mp | |
| (tTensorCoefficient_zero_of_not_dvd q hq hq1 h n hn (aExtKunnethEquiv q n x)) | |
| def aMonomial (n : ℕ) (i : Fin (n+1)) : ASelfExt q (3*n) := | |
| (aExtCoefficientEquiv q hq hq1 h n).symm (Pi.single i 1) | |
| end ArExplicit.Resolution | |
| end | |
| end OAI | |