| import Mathlib |
| import OAI.RingTheory.Multiplicity.NumericalSupport |
| import OAI.RingTheory.Multiplicity.SumEstimate |
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| namespace OAI |
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| noncomputable section |
| namespace Lech |
| open CategoryTheory CategoryTheory.Limits HomologicalComplex ProductSourceCover |
| universe u |
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| 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)) |
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| 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 |
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| end |
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| end OAI |
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