| import OAI.Algebra.AuslanderReiten.Modules.ATwistFunctor |
| import OAI.Algebra.AuslanderReiten.Tensor.TwistedEnvelopingEndpoint |
| import OAI.Algebra.AuslanderReiten.Ext.ExactFunctorExtIso |
| import OAI.Algebra.AuslanderReiten.Tensor.BalancedTensorFunctorExt |
| import OAI.Algebra.AuslanderReiten.Modules.TwistedFiniteEndpoint |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open CategoryTheory CategoryTheory.Abelian |
| open scoped TensorProduct ModuleCat.Algebra |
| namespace ArExplicit.T |
| universe u |
| variable {k : Type u} [Field k] [CharP k 2] (q : k) (H : kˣ) |
| variable [HasDerivedCategory (ModuleCat.{u} (ArExplicit.T k q))] |
| attribute [local instance] moduleCatBaseModule moduleCatBaseTower |
|
|
| abbrev evaluatedTwistFunctor := BalancedTensor.functor (k := k) (R := A k q) (S := A k q) |
| (F := EnvelopingBimodule k (A k q) (twistedFiniteModule (sourceTwist q H)).carrier) |
|
|
| def evaluatedTwistIso := |
| BalancedTensor.nativeEvaluationIso |
| (A.simpleFiniteModule q) (evaluatedTwistEquiv q H) |
|
|
| omit [CharP k 2] [HasDerivedCategory (ModuleCat (T k q))] in |
| theorem evaluatedTwistIso_tmul |
| (a : EnvelopingBimodule k (A k q) (twistedFiniteModule (sourceTwist q H)).carrier) |
| (x : A.Simple q) : |
| type_of% ((BalancedTensor.nativeEvaluationIso_tmul |
| (A.simpleFiniteModule q) (evaluatedTwistEquiv q H) a x).trans |
| (evaluatedTwistEquiv_tmul q H a x)) := |
| (BalancedTensor.nativeEvaluationIso_tmul |
| (A.simpleFiniteModule q) (evaluatedTwistEquiv q H) a x).trans |
| (evaluatedTwistEquiv_tmul q H a x) |
|
|
| omit [CharP k 2] [HasDerivedCategory (ModuleCat (T k q))] in |
| theorem inverseEndpoint_inv_apply (x : (Resolution.aInverseRestriction q H).obj (A.Simple q)) : |
| (Resolution.aInverseEndpoint q H).inv.hom x = x := by |
| have hh := congrArg (fun f => f.hom x) (Resolution.aInverseEndpoint q H).inv_hom_id |
| change (Resolution.aInverseEndpoint q H).hom.hom |
| ((Resolution.aInverseEndpoint q H).inv.hom x) = x at hh |
| rw [Resolution.aInverseEndpoint_apply] at hh |
| exact hh |
|
|
| omit [CharP k 2] [HasDerivedCategory (ModuleCat (T k q))] in |
| theorem evaluatedRestriction_tmul |
| (a : EnvelopingBimodule k (A k q) (twistedFiniteModule (sourceTwist q H)).carrier) |
| (x : A.Simple q) : |
| type_of% ((twistedEnvelopingEndpoint_tmul (sourceTwist q H) (A.Simple q) (A.Simple q) |
| (Resolution.aInverseEndpoint q H).symm a x).trans (inverseEndpoint_inv_apply q H _)) := |
| (twistedEnvelopingEndpoint_tmul (sourceTwist q H) (A.Simple q) (A.Simple q) |
| (Resolution.aInverseEndpoint q H).symm a x).trans (inverseEndpoint_inv_apply q H _) |
|
|
| omit [CharP k 2] [HasDerivedCategory (ModuleCat (T k q))] in |
| theorem evaluatedRestriction_eq : |
| type_of% (twistedFiniteEndpoint_eq (sourceTwist q H) (A.simpleFiniteModule q) |
| (evaluatedTwistEquiv q H) (Resolution.aInverseEndpoint q H) |
| (fun a x => (inverseEndpoint_inv_apply q H _).trans (evaluatedTwistEquiv_tmul q H a x).symm)) := |
| twistedFiniteEndpoint_eq (sourceTwist q H) (A.simpleFiniteModule q) |
| (evaluatedTwistEquiv q H) (Resolution.aInverseEndpoint q H) |
| (fun a x => (inverseEndpoint_inv_apply q H _).trans (evaluatedTwistEquiv_tmul q H a x).symm) |
|
|
| omit [CharP k 2] [HasDerivedCategory (ModuleCat (T k q))] in |
| theorem evaluatedTwistEndpoint_eq : |
| type_of% (twistedFiniteEndpoint_symm_eq (sourceTwist q H) (A.simpleFiniteModule q) |
| (evaluatedTwistEquiv q H) (Resolution.aInverseEndpoint q H) |
| (fun a x => (inverseEndpoint_inv_apply q H _).trans (evaluatedTwistEquiv_tmul q H a x).symm)) := |
| twistedFiniteEndpoint_symm_eq (sourceTwist q H) (A.simpleFiniteModule q) |
| (evaluatedTwistEquiv q H) (Resolution.aInverseEndpoint q H) |
| (fun a x => (inverseEndpoint_inv_apply q H _).trans (evaluatedTwistEquiv_tmul q H a x).symm) |
|
|
| theorem evaluatedTwistExt_scalar (hq : q ≠ 0) (hq1 : 1+q ≠ 0) |
| (hp : Resolution.NoPositivePowerOne q) (n : ℕ) |
| (x : Resolution.ASelfExt q (3*n)) : |
| selfExtIso (k := k) (evaluatedTwistIso q H) (3*n) |
| (x.mapExactFunctor (evaluatedTwistFunctor q H)) = (((H⁻¹ : kˣ) : k)^n) • x := by |
| have he := evaluatedTwistEndpoint_eq q H |
| have hh := ResolutionProducts.exactFunctor_action_natIso (k := k) |
| (evaluatedTwistFunctor q H) (Resolution.aInverseRestriction q H) |
| (twistedEnvelopingRestrictionIso (sourceTwist q H)) |
| (ModuleBar.nativeResolution k (A k q) (A.Simple q)) |
| (Resolution.aInverseEndpoint q H) (3*n) x |
| erw [← he] at hh |
| exact hh.trans (Resolution.aInverseTwist_standard_scalar q H hq hq1 hp n x) |
|
|
| end ArExplicit.T |
|
|
| end |
|
|
| end OAI |
|
|
