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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 | import Mathlib
import OAI.RingTheory.Multiplicity.ComplexEstimate
import OAI.RingTheory.Multiplicity.LocalContractibility
namespace OAI
noncomputable section
namespace Lech
open CategoryTheory CategoryTheory.Limits HomologicalComplex ProductSourceCover
universe u
variable {R : Type u} [CommRing R] {I : Ideal R}
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
def TorsionLength.ZeroStability (ell : TorsionLength I) : Prop :=
ell.DirectSumZero ∧ ell.FilteredColimitZero
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
end
end OAI
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