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2c7d975 | 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 | import OAI.Algebra.AuslanderReiten.Modules.FiniteModuleEvaluation
import OAI.Algebra.AuslanderReiten.Fibers.BimoduleFiber
import OAI.Algebra.AuslanderReiten.Tensor.BalancedTensorEquiv
import OAI.Algebra.AuslanderReiten.Tensor.TwistedTensor
namespace OAI
noncomputable section
namespace ArExplicit
open scoped TensorProduct
universe u
variable {k R : Type u} [Field k] [Ring R] [Algebra k R] [Module.Finite k R]
variable (σ : R ≃ₐ[k] R)
def twistedEvaluationCoordinates :
EnvelopingBimodule k R (twistedFiniteModule σ).carrier ≃ₗ[k] TwistedBimodule k R σ :=
EnvelopingBimodule.scalarEquiv _
theorem twistedEvaluationCoordinates_right (r : Rᵐᵒᵖ)
(x : EnvelopingBimodule k R (twistedFiniteModule σ).carrier) :
twistedEvaluationCoordinates σ (r • x) = r • twistedEvaluationCoordinates σ x := by
change (1 : R) • (r • (show TwistedBimodule k R σ from x.value)) = r • (show TwistedBimodule k R σ from x.value)
rw [one_smul]
theorem twistedEvaluationCoordinates_left (r : R)
(x : EnvelopingBimodule k R (twistedFiniteModule σ).carrier) :
twistedEvaluationCoordinates σ (r • x) = r • twistedEvaluationCoordinates σ x := by
change r • ((1 : Rᵐᵒᵖ) • (show TwistedBimodule k R σ from x.value)) = r • (show TwistedBimodule k R σ from x.value)
rw [one_smul]
namespace FiniteLeftModule
variable (X : FiniteLeftModule k R)
def twistedEvaluationEquiv : (evaluated X (twistedFiniteModule σ)).carrier ≃ₗ[R]
TwistedLeftModule k R σ X.carrier :=
(BalancedTensor.congrLeft (twistedEvaluationCoordinates σ) (LinearEquiv.refl k X.carrier)
(twistedEvaluationCoordinates_right σ) (fun _ _ => rfl)
(twistedEvaluationCoordinates_left σ)).trans (TwistedBimodule.tensorUntwist σ)
@[simp] theorem twistedEvaluationEquiv_tmul
(r : EnvelopingBimodule k R (twistedFiniteModule σ).carrier) (x : X.carrier) :
twistedEvaluationEquiv σ X (BalancedTensor.tmul r x) =
⟨σ.symm r.value.value • x⟩ := rfl
end FiniteLeftModule
end ArExplicit
end
end OAI
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