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d7f9a33 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 | 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
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