| import Mathlib |
| import OAI.RingTheory.Multiplicity.ComplexTheorem |
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| namespace OAI |
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| noncomputable section |
| open CategoryTheory CategoryTheory.Limits HomologicalComplex Lech.ProductSourceCover |
| open scoped ENNReal |
| namespace Lech |
| universe u |
| |
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|
| theorem complex_estimate_allModule (d : ℕ) (hd : 0<d) : |
| ∃ A : ℝ,0≤A ∧ ∀ {C : Type u} [CommRing C] [Nontrivial C] |
| (z : Fin d → C) (ell : AllModuleLength C) (e : ℝ) (_he : 0≤e) |
| (_hmu : ell.value (ModuleCat.of C (C ⧸ Ideal.span (Set.range z))) = ENNReal.ofReal e) |
| (_ha : ∀ a : ℕ,0<a → ell.value (ModuleCat.of C |
| (C ⧸ Ideal.span (Set.range (fun j => z j^a)))) = (a:ℝ≥0∞)^d*ENNReal.ofReal e) |
| (_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} C) ℤ) |
| (_hf : ∀ i,Module.Free C (F.X i)) (_hfin : ∀ i,Module.Finite C (F.X i)) |
| (_hb : ∀ i,i < -(d:ℤ) ∨ 0 < i → IsZero (F.X i)) |
| (_hac : ∀ j,(((baseChangeFunctor C (Localization.Away (z j))).mapHomologicalComplex _).obj F).Acyclic) |
| (s : ℕ) (_hs : 0<s) |
| (_hδ : ∀ i : ℤ,(F.d i (i+1)).hom.range ≤ |
| (Ideal.span (Set.range z))^s • (⊤ : Submodule C (F.X (i+1)))) |
| (_halt : ∑ i∈Finset.range (d+1),(-1:ℝ)^i*(Module.finrank C (F.X (-(i:ℤ))):ℝ)=0), |
| e*(s:ℝ)^d*((Module.finrank C (F.X 0):ℝ)-A/s* |
| ∑ i∈Finset.range (d+1),(Module.finrank C (F.X (-(i:ℤ))):ℝ)) ≤ |
| (ell.value (F.homology 0)).toReal := by |
| classical |
| obtain ⟨n,rfl⟩ := Nat.exists_eq_succ_of_ne_zero hd.ne' |
| 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 C _ _ z ell e he hmu ha hK F hf hfin hb hac s hs hδ halt |
| let I := Ideal.span (Set.range z) |
| let l := ell.torsionLength I |
| let b : ℤ → ℕ := fun i => Module.finrank C (F.X i) |
| let B : ∀ i,Module.Basis (Fin (b i)) C (F.X i) := fun i => by |
| letI := hf i |
| letI := hfin i |
| exact Module.finBasis C (F.X i) |
| have hds : l.DirectSumZero := fun M _ _ _ _ hh => ell.directSum_zero M hh |
| have hlmu : l.value (ModuleCat.of C (C ⧸ I)) ≠ ⊤ := by |
| change ell.value _ ≠ ⊤ |
| rw [hmu] |
| exact ENNReal.ofReal_ne_top |
| have hla (a : ℕ) (ha0 : 0<a) : l.value (ModuleCat.of C |
| (C ⧸ Ideal.span (Set.range (fun j => z j^a)))) = |
| a^(n+1) • l.value (ModuleCat.of C (C ⧸ I)) := by |
| change ell.value _ = a^(n+1) • ell.value _ |
| rw [ha a ha0,hmu] |
| simp only [nsmul_eq_mul,Nat.cast_pow] |
| have H' := (H I z rfl l hds hlmu hla hK F b B (by |
| simpa only [Nat.cast_add,Nat.cast_one,Nat.cast_succ] using hb) hac).2.2 s hs hδ halt |
| change (ell.value (ModuleCat.of C (C ⧸ I))).toReal * _ * _ ≤ _ at H' |
| rw [hmu,ENNReal.toReal_ofReal he] at H' |
| exact H' |
| end Lech |
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| end |
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| end OAI |
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