File size: 2,960 Bytes
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 | import Mathlib
import OAI.RingTheory.Multiplicity.ComplexTheorem
namespace OAI
noncomputable section
open CategoryTheory CategoryTheory.Limits HomologicalComplex Lech.ProductSourceCover
open scoped ENNReal
namespace Lech
universe u
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
end
end OAI
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