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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 | 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
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