openAI-math-dataset / lean /OAI /Algebra /AuslanderReiten /Resolutions /ConcreteNativeResolution.lean
| import OAI.Algebra.AuslanderReiten.Evaluation.ConcreteNativeAugmentation | |
| import OAI.Algebra.AuslanderReiten.Ext.ExternalNativeRegularAugmentation | |
| namespace OAI | |
| noncomputable section | |
| namespace ArExplicit.T | |
| open CategoryTheory | |
| open scoped TensorProduct | |
| private theorem augmentation_compose {C : Type*} [Category C] | |
| {P Q R S U V W : C} (a : P ⟶ Q) (b : Q ⟶ R) (p : R ⟶ S) | |
| (f : Q ⟶ U) (g : U ⟶ S) (u : P ⟶ V) (v : V ⟶ U) | |
| (z : P ⟶ W) (w : W ⟶ S) | |
| (hb : b ≫ p = f ≫ g) (ha : a ≫ f = u ≫ v) (hc : u ≫ (v ≫ g) = z ≫ w) : | |
| (a ≫ b) ≫ p = z ≫ w := by | |
| rw [Category.assoc, hb, ← Category.assoc, ha, Category.assoc, hc] | |
| universe u | |
| variable {k : Type u} [Field k] [CharP k 2] (q : k) | |
| local notation "T₀" => T k q | |
| local notation "A₀" => A k q | |
| local notation "S" => Resolution.Simple q | |
| local notation "X" => A.simpleFiniteModule q | |
| local instance concreteNativeResolutionBaseModule : Module k (canonicalTensorSimple q) := | |
| moduleCatBaseModule (k := k) (R := A₀) (canonicalTensorSimple q) | |
| local instance concreteNativeResolutionBaseTower : IsScalarTower k A₀ (canonicalTensorSimple q) := | |
| moduleCatBaseTower (k := k) (R := A₀) (canonicalTensorSimple q) | |
| def simpleNativeTensorResolution : ProjectiveResolution (canonicalTensorSimple q) := | |
| ExternalTensor.projectiveResolution k T₀ T₀ (ModuleBar.nativeResolution k T₀ S) | |
| (ModuleBar.nativeResolution k T₀ S) | |
| theorem simpleNativeBarIso_augmentation : | |
| (simpleNativeBarIso q).hom.f 0 ≫ (simpleNativeTensorResolution q).π.f 0 = | |
| (simpleOrdinaryResolution q).π.f 0 ≫ | |
| (evaluatedBaseZeroEquiv q).toModuleIso.hom ≫ (nativeToCanonicalSimple q).hom := by | |
| have hbar := ExternalBalanced.nativeTotalBarIso_regularAugmentation k T₀ T₀ S S | |
| have hcoef := (EnvelopingBimodule.evaluationCoefficientIso k A₀ | |
| (nativeToCanonicalSimple q)).hom.naturality (bimoduleAugmentation q) | |
| rw [simpleNativeBarIso_hom_f] | |
| erw [simpleOrdinaryBarIso_f, Category.id_comp] | |
| exact augmentation_compose | |
| ((simpleCoefficientComplexIso q).hom.f 0) | |
| ((ExternalBalanced.nativeTotalBarIso k T₀ T₀ S S).hom.f 0) | |
| ((simpleNativeTensorResolution q).π.f 0) | |
| ((EnvelopingBimodule.evaluationFunctor k A₀ (canonicalTensorSimple q)).map (bimoduleAugmentation q)) | |
| (BarBalanced.nativeEndpointIso k A₀ (canonicalTensorSimple q)).hom | |
| ((EnvelopingBimodule.evaluationFunctor k A₀ (X).carrier).map (bimoduleAugmentation q)) | |
| ((EnvelopingBimodule.evaluationCoefficientIso k A₀ (nativeToCanonicalSimple q)).hom.app | |
| (AbsoluteBar.moduleObject k A₀ 0)) | |
| ((simpleOrdinaryResolution q).π.f 0) | |
| ((evaluatedBaseZeroEquiv q).toModuleIso.hom ≫ (nativeToCanonicalSimple q).hom) | |
| hbar hcoef.symm (simpleBaseAugmentation q) | |
| def simpleNativeResolutionHom : ProjectiveResolution.Hom (simpleOrdinaryResolution q) | |
| (simpleNativeTensorResolution q) | |
| ((evaluatedBaseZeroEquiv q).toModuleIso.hom ≫ (nativeToCanonicalSimple q).hom) where | |
| hom := (simpleNativeBarIso q).hom | |
| hom_f_zero_comp_π_f_zero := simpleNativeBarIso_augmentation q | |
| end ArExplicit.T | |
| end | |
| end OAI | |