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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 | import Mathlib
import OAI.RingTheory.Multiplicity.NumericalSupport
import OAI.RingTheory.Multiplicity.SumEstimate
namespace OAI
noncomputable section
namespace Lech
open CategoryTheory CategoryTheory.Limits HomologicalComplex ProductSourceCover
universe u
lemma rank_zero_reverse (h : ℕ) (b : ℤ → ℕ)
(hb : ∑ i∈Finset.range (h+1),(-1:ℝ)^i*(b (-(i:ℤ)):ℝ)=0) :
∑ k∈Finset.range (h+1),(-1:ℝ)^k*(b (-(h:ℤ)+k):ℝ)=0 := by
have he := signed_reverse h (fun p => (b p:ℝ)) (fun _=>1)
simp only [mul_one,hb] at he
exact (mul_eq_zero.mp he).resolve_left (pow_ne_zero _ (by norm_num))
theorem complex_estimate_basis (n : ℕ) [LinearOrder (Chart n)] :
∃ A : ℝ,0≤A ∧ ∀ {R : Type u} [CommRing R] [Nontrivial R]
(I : Ideal R) (z : Fin (n+1) → R)
(_hgen : Ideal.span (Set.range z)=I)
(ell : TorsionLength I) (_hds : ell.DirectSumZero)
(_hmu : ell.value (ModuleCat.of R (R ⧸ I))≠⊤)
(_ha : ∀ a : ℕ,0<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)
(b : ℤ → ℕ) (_B : ∀ p,Module.Basis (Fin (b p)) R (F.X p))
(_hb : ∀ p,p < -(n+1:ℤ) ∨ 0<p → IsZero (F.X p))
(_hac : ∀ k,((baseChangeFunctor R (Localization.Away (z k))).mapHomologicalComplex _ |>.obj F).Acyclic),
(∀ i,powerTorsion I (F.homology i) ∧ ell.value (F.homology i)≠⊤) ∧
(∀ i : ℤ,i≠0 → ell.value (F.homology i)=0) ∧
∀ s : ℕ,0<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*(b (-(i:ℤ)):ℝ)=0) →
ell.realValue (ModuleCat.of R (R ⧸ I))*(s:ℝ)^(n+1)*
((b 0:ℝ)-A/s*∑ i∈Finset.range (n+2),(b (-(i:ℤ)):ℝ))≤
ell.realValue (F.homology 0) := by
obtain ⟨A,hA,hest⟩ := Numerical.uniform_from_sum_estimate n
refine ⟨A,hA,?_⟩
intro R _ _ I z hgen ell hds hmu ha hK F b B hb hac
have hf : ∀ p,Module.Free R (F.X p) := fun p => Module.Free.of_basis (B p)
have hfin : ∀ p,Module.Finite R (F.X p) := fun p => Module.Finite.of_basis (B p)
have hz : ∀ j,z j∈I := fun j => hgen ▸ Ideal.subset_span ⟨j,rfl⟩
have hbf : ∀ p,p < -((n+1:ℕ):ℤ) ∨ 0<p → IsZero (F.X p) := by
simpa only [Nat.cast_add,Nat.cast_one] using hb
have hpure := SourceGraded.acyclicity_finite_torsion I z hgen ell F hmu ha hK hf hfin hbf hac
refine ⟨hpure.1,hpure.2,?_⟩
intro s hs hd halt
have hbr := rank_zero_reverse (n+1) b halt
simp only [Nat.cast_add,Nat.cast_one] at hbr
have H := hest s hs (fun i => (b (-(i.val:ℤ)):ℝ)) (fun _ => Nat.cast_nonneg _)
(ell.realValue (ModuleCat.of R (R ⧸ I))) (ell.realValue (F.homology 0))
ENNReal.toReal_nonneg ENNReal.toReal_nonneg
have halt' : (∑ i : Fin (n+2),(-1:ℝ)^i.val*(b (-(i.val:ℤ)):ℝ))=0 := by
rw [Fin.sum_univ_eq_sum_range (fun i => (-1:ℝ)^i*(b (-(i:ℤ)):ℝ)) (n+2)]
exact halt
have he : (∑ i : Fin (n+2),(b (-(i.val:ℤ)):ℝ))=
∑ i∈Finset.range (n+2),(b (-(i:ℤ)):ℝ) :=
Fin.sum_univ_eq_sum_range (fun i => (b (-(i:ℤ)):ℝ)) (n+2)
apply (show _ from H halt' ?_) |> fun hh => by simpa only [Fin.val_zero,Nat.cast_zero,neg_zero,he] using hh
intro hr
have hr' : RootPositions n s (Numerical.roundedRoot n s) := by
intro j
constructor
· exact_mod_cast (hr j).1
· exact_mod_cast (hr j).2
have ht := ReesRoot.sum_estimate I z hz hgen ell hds hmu ha hK F s hd b B hf hfin hb hac hbr
(Numerical.roundedRoot n s) hr'
rw [Fin.sum_univ_eq_sum_range (fun i => (-1:ℝ)^i*(b (-(i:ℤ)):ℝ)*Numerical.layer n i s) (n+2)]
exact ht
end Lech
end
end OAI
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