| 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 |
|
|
