| import Mathlib |
| import OAI.RingTheory.Multiplicity.ComplexEstimate |
| import OAI.RingTheory.Multiplicity.LocalContractibility |
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| namespace OAI |
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| noncomputable section |
| namespace Lech |
| open CategoryTheory CategoryTheory.Limits HomologicalComplex ProductSourceCover |
| universe u |
| variable {R : Type u} [CommRing R] {I : Ideal R} |
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| def TorsionLength.FilteredColimitZero (ell : TorsionLength I) : Prop := |
| ∀ {J : Type u} [SmallCategory J] [IsFiltered J] (D : J ⥤ ModuleCat.{u} R) |
| (a : ℕ), (∀ j,I^a ≤ Module.annihilator R (D.obj j)) → |
| (∀ j,ell.value (D.obj j)=0) → ell.value (colimit D)=0 |
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| def TorsionLength.ZeroStability (ell : TorsionLength I) : Prop := |
| ell.DirectSumZero ∧ ell.FilteredColimitZero |
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| theorem complex_theorem (n : ℕ) : |
| ∃ A : ℝ,0≤A ∧ ∀ {R : Type u} [CommRing R] |
| (I : Ideal R) (z : Fin (n+1) → R) |
| (_hgen : Ideal.span (Set.range z)=I) |
| (ell : TorsionLength I) (_hst : ell.ZeroStability) |
| (_hmupos : 0<ell.value (ModuleCat.of R (R ⧸ I))) |
| (_hmu : ell.value (ModuleCat.of R (R ⧸ I))≠⊤) |
| (_ha : ∀ a : ℕ,1≤a → ell.value (ModuleCat.of R |
| (R ⧸ Ideal.span (Set.range (fun j => z j^a))))= |
| a^(n+1) • ell.value (ModuleCat.of R (R ⧸ I))) |
| (_hK : ∀ a : ℕ,1≤a → ∀ i : ℤ,i<0 → |
| ell.value ((Koszul.unit (List.ofFn (fun j => z j^a))).homology i)=0) |
| (F : CochainComplex (ModuleCat.{u} R) ℤ) |
| (_hf : ∀ p,Module.Free R (F.X p)) (_hfin : ∀ p,Module.Finite R (F.X p)) |
| (_hb : ∀ p,p < -(n+1:ℤ) ∨ 0<p → IsZero (F.X p)) |
| (_hloc : LocallyContractibleOff I F), |
| (∀ i,powerTorsion I (F.homology i) ∧ ell.value (F.homology i)≠⊤) ∧ |
| (∀ i : ℤ,i≠0 → ell.value (F.homology i)=0) ∧ |
| ∀ s : ℕ, 1 ≤ s → |
| (∀ p : ℤ,(F.d p (p+1)).hom.range ≤ I^s • (⊤ : Submodule R (F.X (p+1)))) → |
| (∑ i∈Finset.range (n+2),(-1:ℝ)^i*(Module.finrank R (F.X (-(i:ℤ))):ℝ)=0) → |
| ell.realValue (ModuleCat.of R (R ⧸ I))*(s:ℝ)^(n+1)* |
| ((Module.finrank R (F.X 0):ℝ)-A/s* |
| ∑ i∈Finset.range (n+2),(Module.finrank R (F.X (-(i:ℤ))):ℝ))≤ |
| ell.realValue (F.homology 0) := by |
| classical |
| let : LinearOrder (Chart n) := LinearOrder.lift' (Fintype.equivFin (Chart n)) |
| (Fintype.equivFin (Chart n)).injective |
| obtain ⟨A,hA,H⟩ := complex_estimate_basis n |
| refine ⟨A,hA,?_⟩ |
| intro R _ I z hgen ell hst hmupos hmu ha hK F hf hfin hb hloc |
| rcases subsingleton_or_nontrivial R with hr|hr |
| · let := hr |
| have he : ell.value (ModuleCat.of R (R ⧸ I))=0 := |
| ell.zero (ModuleCat.isZero_iff_subsingleton.mpr inferInstance) |
| exact False.elim ((ne_of_gt hmupos) he) |
| · let := hr |
| let b : ℤ → ℕ := fun p => Module.finrank R (F.X p) |
| let B : ∀ p,Module.Basis (Fin (b p)) R (F.X p) := fun p => by |
| letI := hf p |
| letI := hfin p |
| exact Module.finBasis R (F.X p) |
| have hac : ∀ k,((baseChangeFunctor R (Localization.Away (z k))).mapHomologicalComplex _ |>.obj F).Acyclic := |
| fun k => hloc.acyclicAway (z k) (hgen ▸ Ideal.subset_span ⟨k,rfl⟩) |
| obtain ⟨ht,hzero,hest⟩ := H I z hgen ell hst.1 hmu (fun a ha0 => ha a (by omega)) hK F b B hb hac |
| exact ⟨ht,hzero,fun s hs hd halt => hest s (by omega) hd halt⟩ |
| end Lech |
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| end |
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| end OAI |
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