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e3e0a0d | 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 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 | 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
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