| import OAI.Algebra.AuslanderReiten.Tensor.BalancedTensorMapEquiv |
| import OAI.Algebra.AuslanderReiten.Tensor.ExternalTensorScalar |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open CategoryTheory |
| namespace ArExplicit.BalancedTensor |
| universe u |
| variable {k R S F : Type u} [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 (M : Type u) [AddCommGroup M] [Module R M] [mM : Module k M] [IsScalarTower k R M] |
|
|
| theorem nativeObject_eq : |
| (functor (k := k) (R := R) (S := S) (F := F)).obj (ModuleCat.of R M) = |
| ModuleCat.of S (BalancedTensor k R F M) := by |
| cases ExternalTensor.canonicalScalar_eq k R M (mM := mM) |
| rfl |
|
|
| def nativeObjectIso : |
| (functor (k := k) (R := R) (S := S) (F := F)).obj (ModuleCat.of R M) ≅ |
| ModuleCat.of S (BalancedTensor k R F M) := |
| eqToIso (nativeObject_eq (k := k) (R := R) (S := S) (F := F) M) |
|
|
| theorem nativeObjectIso_tmul (f : F) (m : M) : |
| (nativeObjectIso (k := k) (R := R) (S := S) (F := F) M).hom.hom |
| (letI := moduleCatBaseModule (k := k) (R := R) (ModuleCat.of R M) |
| tmul f m) = tmul f m := by |
| cases ExternalTensor.canonicalScalar_eq k R M (mM := mM) |
| rfl |
|
|
| variable {M} |
| variable {L : Type u} [AddCommGroup L] [Module R L] [mL : Module k L] [IsScalarTower k R L] |
|
|
| theorem nativeObjectIso_natural (f : M →ₗ[R] L) : |
| (functor (k := k) (R := R) (S := S) (F := F)).map (ModuleCat.ofHom f) ≫ |
| (nativeObjectIso (k := k) (R := R) (S := S) (F := F) L).hom = |
| (nativeObjectIso (k := k) (R := R) (S := S) (F := F) M).hom ≫ |
| ModuleCat.ofHom (mapRight (k := k) (F := F) (S := S) f) := by |
| cases ExternalTensor.canonicalScalar_eq k R M (mM := mM) |
| cases ExternalTensor.canonicalScalar_eq k R L (mM := mL) |
| rfl |
|
|
| end ArExplicit.BalancedTensor |
|
|
| end |
|
|
| end OAI |
|
|
