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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 | import OAI.Algebra.AuslanderReiten.Tensor.BalancedTensorCategory
namespace OAI
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]
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
@[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
end ArExplicit.BalancedTensor
end
end OAI
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