| import OAI.Algebra.AuslanderReiten.Tensor.BalancedTensorCategory |
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| namespace OAI |
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| noncomputable section |
| namespace ArExplicit.BalancedTensor |
| variable {k R S F L M : Type*} [Field k] [Ring R] [Ring S] [Algebra k R] |
| variable [AddCommGroup F] [Module k F] [Module Rᵐᵒᵖ F] [Module S F] |
| variable [SMulCommClass k S F] [SMulCommClass S Rᵐᵒᵖ F] |
| variable [AddCommGroup L] [Module k L] [Module R L] [IsScalarTower k R L] |
| variable [AddCommGroup M] [Module k M] [Module R M] [IsScalarTower k R M] |
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| def mapRightEquiv (e : L ≃ₗ[R] M) : BalancedTensor k R F L ≃ₗ[S] BalancedTensor k R F M where |
| __ := mapRight (k := k) (F := F) (S := S) e.toLinearMap |
| invFun := mapRight (k := k) (F := F) (S := S) e.symm.toLinearMap |
| left_inv x := by |
| change mapRight (k := k) (F := F) (S := S) e.symm.toLinearMap |
| (mapRight (k := k) (F := F) (S := S) e.toLinearMap x) = x |
| have h := congrArg (fun t : BalancedTensor k R F L →ₗ[S] BalancedTensor k R F L => t x) |
| (mapRight_comp (k := k) (F := F) (S := S) e.toLinearMap e.symm.toLinearMap) |
| simpa only [e.symm_comp, mapRight_id, LinearMap.comp_apply, LinearMap.id_apply] using h.symm |
| right_inv x := by |
| change mapRight (k := k) (F := F) (S := S) e.toLinearMap |
| (mapRight (k := k) (F := F) (S := S) e.symm.toLinearMap x) = x |
| have h := congrArg (fun t : BalancedTensor k R F M →ₗ[S] BalancedTensor k R F M => t x) |
| (mapRight_comp (k := k) (F := F) (S := S) e.symm.toLinearMap e.toLinearMap) |
| simpa only [e.comp_symm, mapRight_id, LinearMap.comp_apply, LinearMap.id_apply] using h.symm |
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| @[simp] theorem mapRightEquiv_tmul (e : L ≃ₗ[R] M) (f : F) (x : L) : |
| mapRightEquiv (k := k) (F := F) (S := S) e (tmul f x) = tmul f (e x) := rfl |
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| end ArExplicit.BalancedTensor |
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| end |
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| end OAI |
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