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b797a5b | 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 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 | import OAI.NumberTheory.PiExponent.Ampleness.ReesExceptional
import OAI.NumberTheory.PiExponent.Jets.CompactJetIdealCoherent
import OAI.NumberTheory.PiExponent.Jets.CompactJetPolynomial
namespace OAI
noncomputable section
open CategoryTheory AlgebraicGeometry
open scoped BigOperators
namespace PiExponent.CompactLogJetIdeal
open CompactJetPolynomial
variable {m : ℕ}
abbrev coordinateRing (m : ℕ) := MvPolynomial (Fin (m+1)) ℂ
abbrev affineSpace (m : ℕ) : Scheme := Spec (CommRingCat.of (coordinateRing m))
abbrev base : Scheme := Spec (CommRingCat.of ℂ)
def structureMap (m : ℕ) : affineSpace m ⟶ base :=
Spec.map (CommRingCat.ofHom (algebraMap ℂ (coordinateRing m)))
def point (c : Fin m → ℂ) : base ⟶ affineSpace m :=
Spec.map (CommRingCat.ofHom (MvPolynomial.aeval (center c)).toRingHom)
theorem point_section (c : Fin m → ℂ) : point c ≫ structureMap m = 𝟙 base := by
change Spec.map _ ≫ Spec.map _ = _
rw [← Spec.map_comp]
have h : CommRingCat.ofHom (algebraMap ℂ (coordinateRing m)) ≫
CommRingCat.ofHom (MvPolynomial.aeval (center c)).toRingHom =
𝟙 (CommRingCat.of ℂ) := by
ext a
simp
rw [h, Spec.map_id]
theorem range_point (c : Fin m → ℂ) : Set.range (point c) = {centerPoint c} := by
change Set.range (PrimeSpectrum.comap (MvPolynomial.aeval (center c)).toRingHom) = _
rw [range_comap_of_surjective _ _ (by
intro a
exact ⟨MvPolynomial.C a, by simp⟩)]
exact PrimeSpectrum.zeroLocus_eq_singleton (WeightedBezout.pointIdeal (center c))
theorem support_specIdeal {R : Type} [CommRing R] (I : Ideal R) :
((PiExponentSeshadri.IdealPullback.specIdeal I).support : Set (Spec (CommRingCat.of R))) =
PrimeSpectrum.zeroLocus (I : Set R) := by
rw [PiExponentSeshadri.IdealPullback.specIdeal,
Scheme.IdealSheafData.coe_support_ofIdealTop, Spec_zeroLocus, ← Ideal.coe_comap]
rw [Ideal.comap_map_of_bijective _
(ConcreteCategory.bijective_of_isIso (Scheme.ΓSpecIso (CommRingCat.of R)).inv)]
def polynomialIdeal {J : Type*} [Fintype J] (c : J → Fin m → ℂ)
(T : Fin m → ℕ) (e : Fin (m+1) → ℕ) : Ideal (coordinateRing m) :=
∏ a, powerIdeal (c a) (logPolynomials T) e
def affineIdeal {J : Type*} [Fintype J] (c : J → Fin m → ℂ)
(T : Fin m → ℕ) (e : Fin (m+1) → ℕ) : (affineSpace m).IdealSheafData :=
PiExponentSeshadri.IdealPullback.specIdeal (polynomialIdeal c T e)
theorem support_affineIdeal {J : Type*} [Fintype J] (c : J → Fin m → ℂ)
(T : Fin m → ℕ) (e : Fin (m+1) → ℕ) (he : ∀ i, 0 < e i) :
((affineIdeal c T e).support : Set (affineSpace m)) =
⋃ a, Set.range (point (c a)) := by
rw [affineIdeal, support_specIdeal]
unfold polynomialIdeal
rw [zeroLocus_prod_powerIdeal c (logPolynomials T) (logPolynomials_eval_zero T) e he]
simp only [range_point]
ext p
constructor
· rintro ⟨a, rfl⟩
exact Set.mem_iUnion.mpr ⟨a, Set.mem_singleton _⟩
· intro hp
obtain ⟨a, ha⟩ := Set.mem_iUnion.mp hp
exact ⟨a, (Set.mem_singleton_iff.mp ha).symm⟩
def compactIdeal {J : Type*} [Fintype J] (c : J → Fin m → ℂ)
(T : Fin m → ℕ) (e : Fin (m+1) → ℕ)
{X : Scheme} (j : affineSpace m ⟶ X) : X.IdealSheafData :=
CompactJetIdeal.extend (affineIdeal c T e) j
theorem restrict_compactIdeal {J : Type*} [Fintype J] (c : J → Fin m → ℂ)
(T : Fin m → ℕ) (e : Fin (m+1) → ℕ)
{X : Scheme} (j : affineSpace m ⟶ X) [IsOpenImmersion j] [QuasiCompact j] :
(compactIdeal c T e j).comap j = affineIdeal c T e :=
CompactJetIdeal.restrict_extend _ j
theorem support_compactIdeal {J : Type*} [Fintype J] (c : J → Fin m → ℂ)
(T : Fin m → ℕ) (e : Fin (m+1) → ℕ) (he : ∀ i, 0 < e i)
{X : Scheme} (j : affineSpace m ⟶ X) [QuasiCompact j]
(π : X ⟶ base) [IsSeparated π] (hπ : j ≫ π = structureMap m) :
((compactIdeal c T e j).support : Set X) =
⋃ a, Set.range (point (c a) ≫ j) := by
rw [compactIdeal, CompactJetIdeal.support_extend (affineIdeal c T e) j π
(fun a => point (c a)) (fun a => by
rw [Category.assoc, hπ, point_section]) (support_affineIdeal c T e he),
support_affineIdeal c T e he, Set.image_iUnion]
congr 1
funext a
rw [Scheme.Hom.comp_base, TopCat.coe_comp, Set.range_comp]
theorem compactIdeal_isFinitePresentation {J : Type*} [Fintype J]
(c : J → Fin m → ℂ) (T : Fin m → ℕ) (e : Fin (m+1) → ℕ)
{X : Scheme} [IsLocallyNoetherian X] (j : affineSpace m ⟶ X) :
(PiExponentSeshadri.IdealModule.closedModule (compactIdeal c T e j)).IsFinitePresentation :=
CompactJetIdeal.extend_isFinitePresentation _ j
theorem polynomialIdeal_pow_le {J : Type*} [Fintype J]
(c : J → Fin m → ℂ) (T : Fin m → ℕ) (e : Fin (m+1) → ℕ) (n : ℕ) (a : J) :
polynomialIdeal c T e ^ n ≤ powerIdeal (c a) (logPolynomials T) e ^ n := by
classical
apply pow_le_pow_left'
exact Ideal.prod_le_inf.trans (Finset.inf_le (Finset.mem_univ a))
theorem formalJet_packet_zero_of_mem_polynomialIdeal_pow {J : Type*} [Fintype J]
(c : J → Fin m → ℂ) (T : Fin m → ℕ) (e : Fin (m+1) → ℕ)
(v : Fin (m+1) → ℚ) (hv : ∀ i, 0 ≤ v i)
(hT : ∀ i, v i.succ ≤ (T i : ℚ) * v 0)
(R : ℚ) (he : ∀ i, R ≤ (e i : ℚ) * v i) (n : ℕ)
(P : coordinateRing m) (hP : P ∈ polynomialIdeal c T e ^ n) (a : J) :
JetGeometry.rationalCoefficientPacket v (n * R) (FormalLogJet.formalJet (c a) P) =
(fun _ => (0 : ℂ)) :=
formalJet_packet_zero_of_mem_pow (c a) T e v hv hT R he n P
(polynomialIdeal_pow_le c T e n a hP)
end PiExponent.CompactLogJetIdeal
end
end OAI
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