| 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 | |