| import Mathlib | |
| import OAI.RingTheory.Multiplicity.ScalarExtensionFiniteHomology | |
| namespace OAI | |
| noncomputable section | |
| open CategoryTheory CategoryTheory.Limits HomologicalComplex MonoidalCategory | |
| open scoped TensorProduct | |
| namespace Lech | |
| universe u | |
| variable {R : Type u} [CommRing R] | |
| def tensorModuleFunctor (F : CochainComplex (ModuleCat.{u} R) ℤ) : | |
| ModuleCat.{u} R ⥤ CochainComplex (ModuleCat.{u} R) ℤ where | |
| obj M := (TensorTotal.termTensor M).obj F | |
| map f := TensorTotal.termTensorMap F _ f | |
| map_id M := by | |
| ext i : 1 | |
| exact whiskerLeft_id (F.X i) M | |
| map_comp f g := TensorTotal.termTensorMap_comp F f g | |
| instance tensorModuleFunctor_additive (F : CochainComplex (ModuleCat.{u} R) ℤ) : | |
| (tensorModuleFunctor F).Additive where | |
| map_add := by | |
| intros M N f g | |
| ext i : 1 | |
| exact Functor.map_add ((curriedTensor (ModuleCat R)).obj (F.X i)) | |
| lemma tensorModuleFunctor_smul (F : CochainComplex (ModuleCat.{u} R) ℤ) | |
| {M N : ModuleCat.{u} R} (a : R) (f : M ⟶ N) : | |
| (tensorModuleFunctor F).map (a • f) = a • (tensorModuleFunctor F).map f := by | |
| ext i : 1 | |
| apply ModuleCat.hom_ext | |
| apply TensorProduct.ext | |
| ext x y | |
| change x ⊗ₜ[R] (a • f y) = a • (x ⊗ₜ[R] f y) | |
| exact TensorProduct.tmul_smul _ _ _ | |
| def tensorModuleUnitIso (F : CochainComplex (ModuleCat.{u} R) ℤ) : | |
| (tensorModuleFunctor F).obj (ModuleCat.of R R) ≅ F := | |
| Hom.isoOfComponents (fun i => (TensorProduct.rid R (F.X i)).toModuleIso) (by | |
| intro i j _ | |
| apply ModuleCat.hom_ext | |
| apply TensorProduct.ext | |
| ext x | |
| change F.d i j ((1:R) • x) = (1:R) • F.d i j x | |
| exact map_smul _ _ _) | |
| lemma tensorModule_homology_length_free [Nontrivial R] | |
| (F : CochainComplex (ModuleCat.{u} R) ℤ) (M : ModuleCat.{u} R) | |
| [Module.Free R M] [Module.Finite R M] (i : ℤ) : | |
| Module.length R (((tensorModuleFunctor F).obj M).homology i) = | |
| Module.finrank R M • Module.length R (F.homology i) := by | |
| change Module.length R (((tensorModuleFunctor F) ⋙ homologyFunctor (ModuleCat R) (.up ℤ) i).obj M) = _ | |
| rw [additive_length_finite_free | |
| ((tensorModuleFunctor F) ⋙ homologyFunctor (ModuleCat R) (.up ℤ) i) M] | |
| congr 1 | |
| exact ((homologyFunctor (ModuleCat R) (.up ℤ) i).mapIso (tensorModuleUnitIso F)).toLinearEquiv.length_eq | |
| theorem exists_tensor_homology_bound [IsDomain R] [IsNoetherianRing R] | |
| (M : ModuleCat.{u} R) [Module.Finite R M] : | |
| ∃ (r : ℕ) (g : R), g ≠ 0 ∧ ∀ (F : CochainComplex (ModuleCat.{u} R) ℤ) (i : ℤ), | |
| (∀ j,Function.Injective (fun x : ((tensorModuleFunctor F).obj M).X j => g • x)) → | |
| Module.length R (((tensorModuleFunctor F).obj M).homology i) ≠ ⊤ → | |
| Module.length R (((tensorModuleFunctor F).obj M).homology i) ≤ | |
| r • Module.length R (F.homology i) + | |
| Module.length R (((complexQuotient (Ideal.span {g}) (.up ℤ)).obj | |
| ((tensorModuleFunctor F).obj M)).homology i) := by | |
| obtain ⟨r,g,hg,u,v,huv,hvu⟩ := finite_free_factorization R M | |
| refine ⟨r,g,hg,fun F i hgi hfin => ?_⟩ | |
| let U : ModuleCat.of R (Fin r → R) ⟶ M := ModuleCat.ofHom u | |
| let V : M ⟶ ModuleCat.of R (Fin r → R) := ModuleCat.ofHom v | |
| have hVU : V ≫ U = g • 𝟙 M := ModuleCat.hom_ext huv | |
| have he := congrArg (tensorModuleFunctor F).map hVU | |
| rw [CategoryTheory.Functor.map_comp,tensorModuleFunctor_smul, | |
| CategoryTheory.Functor.map_id] at he | |
| have hb := homology_length_le_of_scalar_retract | |
| ((tensorModuleFunctor F).map V) ((tensorModuleFunctor F).map U) g he i hgi hfin | |
| rw [tensorModule_homology_length_free F (ModuleCat.of R (Fin r → R)) i] at hb | |
| simpa only [Module.finrank_fintype_fun_eq_card,Fintype.card_fin] using hb | |
| end Lech | |
| end | |
| end OAI | |