| import OAI.Algebra.AuslanderReiten.Ext.ExternalFreeEvaluation |
| |
| namespace OAI |
|
|
| noncomputable section |
| namespace ArExplicit.ExternalBalanced |
| open CategoryTheory |
| open scoped TensorProduct |
| private theorem replace_product {C : Type*} [Category C] |
| {P Q R S T : C} (d : P ⟶ Q) (p : Q ⟶ R) (q : R ⟶ S) |
| (r : S ⟶ T) (s : Q ⟶ S) (h : s = p ≫ q) : |
| d ≫ p ≫ q ≫ r = d ≫ s ≫ r := by |
| rw [h, Category.assoc] |
|
|
| 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) |
| variable (U : ModuleCat.{u} A) (V : ModuleCat.{u} B) |
| variable (a : (EnvelopingBimodule.evaluationFunctor k A X).obj M ⟶ U) |
| (b : (EnvelopingBimodule.evaluationFunctor k B Y).obj N ⟶ V) |
| variable (f : U ⟶ ModuleCat.of A A) (g : V ⟶ ModuleCat.of B B) |
| |
| theorem freeEvaluation_factorization : |
| (shuffle k A B M N X Y).toModuleIso.hom ≫ ExternalTensor.mapHom k A B a b ≫ |
| ExternalTensor.mapHom k A B f g ≫ |
| (ExternalTensor.canonicalTensorIso k A B A B).hom ≫ |
| (ExternalTensor.regularEquiv k A B).toModuleIso.hom = |
| freeEvaluation k A B M N X Y (a ≫ f) (b ≫ g) := |
| replace_product (shuffle k A B M N X Y).toModuleIso.hom |
| (ExternalTensor.mapHom k A B a b) (ExternalTensor.mapHom k A B f g) |
| ((ExternalTensor.canonicalTensorIso k A B A B).hom ≫ |
| (ExternalTensor.regularEquiv k A B).toModuleIso.hom) |
| (ExternalTensor.mapHom k A B (a ≫ f) (b ≫ g)) |
| (ExternalTensor.mapHom_comp k A B a f b g) |
|
|
| end ArExplicit.ExternalBalanced |
| |
| end |
|
|
| end OAI |
|
|
