File size: 1,512 Bytes
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 | import OAI.Algebra.AuslanderReiten.Ext.ExternalEvaluationTerms
import OAI.Algebra.AuslanderReiten.Tensor.BalancedFourAddExt
namespace OAI
noncomputable section
namespace ArExplicit.ExternalBalanced
open CategoryTheory
open scoped TensorProduct
universe u
variable (k A B : Type u) [Field k] [Ring A] [Ring B] [Algebra k A] [Algebra k B]
attribute [local instance 2000] moduleCatBaseModule moduleCatBaseTower
variable (M : ModuleCat.{u} (A ⊗[k] Aᵐᵒᵖ)) (N : ModuleCat.{u} (B ⊗[k] Bᵐᵒᵖ))
variable (X : ModuleCat.{u} A) (Y : ModuleCat.{u} B) (Z : ModuleCat.{u} (A ⊗[k] B))
theorem exterior_hom_ext
(f g : (productEvaluation k A B X Y).obj
(ExternalTensor.moduleObject k (A ⊗[k] Aᵐᵒᵖ) (B ⊗[k] Bᵐᵒᵖ) M N) ⟶ Z)
(h : ∀ (a : EnvelopingBimodule k A M) (b : EnvelopingBimodule k B N) (x : X) (y : Y),
f (BalancedTensor.tmul ((bimoduleCoordinates k A B M N).symm (a ⊗ₜ[k] b))
((ExternalTensor.moduleScalarCoordinates k A B X Y).symm (x ⊗ₜ[k] y))) =
g (BalancedTensor.tmul ((bimoduleCoordinates k A B M N).symm (a ⊗ₜ[k] b))
((ExternalTensor.moduleScalarCoordinates k A B X Y).symm (x ⊗ₜ[k] y)))) : f = g := by
apply ModuleCat.hom_ext
apply LinearMap.toAddMonoidHom_injective
exact fourfold_add_hom_ext (k := k) (A := A) (B := B) (Z := Z)
(bimoduleCoordinates k A B M N) (ExternalTensor.moduleScalarCoordinates k A B X Y)
f.hom.toAddMonoidHom g.hom.toAddMonoidHom h
end ArExplicit.ExternalBalanced
end
end OAI
|