File size: 2,736 Bytes
a72760f | 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 | import Mathlib.AlgebraicGeometry.IdealSheaf.Basic
import Mathlib.AlgebraicGeometry.Noetherian
import Mathlib.RingTheory.Noetherian.Nilpotent
namespace OAI
namespace PiExponentSeshadri.Geometry
noncomputable section
open AlgebraicGeometry CategoryTheory TopologicalSpace
variable {X : Scheme}
lemma idealSheaf_le_nilradical_of_fullSupport (I : X.IdealSheafData)
(hI : I.support = ⊤) : I ≤ X.nilradical := by
have h : (⊤ : Closeds X) ≤ I.support := by rw [hI]
simpa only [Scheme.IdealSheafData.vanishingIdeal_top] using
Scheme.IdealSheafData.le_support_iff_le_vanishingIdeal.mp h
lemma affineIdeal_isNilpotent_of_fullSupport [IsLocallyNoetherian X]
(I : X.IdealSheafData) (hI : I.support = ⊤) (U : X.affineOpens) :
IsNilpotent (I.ideal U) := by
let := IsLocallyNoetherian.component_noetherian U
apply (Ideal.FG.isNilpotent_iff_le_nilradical (IsNoetherian.noetherian _)).mpr
exact idealSheaf_le_nilradical_of_fullSupport I hI U
theorem exists_pos_pow_eq_bot_of_fullSupport [IsNoetherian X]
(I : X.IdealSheafData) (hI : I.support = ⊤) :
∃ n : ℕ, 0 < n ∧ I ^ n = ⊥ := by
classical
have hlocal : ∀ U : X.affineOpens, ∃ n : ℕ, I.ideal U ^ n = ⊥ :=
fun U => affineIdeal_isNilpotent_of_fullSupport I hI U
choose n hn using hlocal
obtain ⟨S, hS⟩ := isCompact_univ.elim_finite_subcover
(fun U : X.affineOpens => (U.1 : Set X)) (fun U => U.1.isOpen)
(by
intro x _
have hx : x ∈ (⨆ U : X.affineOpens, U.1) := by
rw [iSup_affineOpens_eq_top X]
trivial
obtain ⟨U, hU⟩ := Opens.mem_iSup.mp hx
exact Set.mem_iUnion.mpr ⟨U, hU⟩)
refine ⟨S.sup n + 1, Nat.succ_pos _, ?_⟩
apply Scheme.IdealSheafData.ext_of_iSup_eq_top (fun U : S => U.val)
· apply top_unique
intro x _
obtain ⟨U, hU⟩ := Set.mem_iUnion.mp (hS (show x ∈ Set.univ from trivial))
obtain ⟨hUS, hxU⟩ := Set.mem_iUnion.mp hU
exact Opens.mem_iSup.mpr ⟨⟨U, hUS⟩, hxU⟩
· intro U
change I.ideal U.val ^ (S.sup n + 1) = ⊥
apply le_bot_iff.mp
calc
I.ideal U.val ^ (S.sup n + 1) ≤ I.ideal U.val ^ n U.val :=
Ideal.pow_le_pow_right ((Finset.le_sup U.property).trans (Nat.le_succ _))
_ = ⊥ := hn U.val
theorem exists_pos_pow_eq_bot_of_support_univ [IsNoetherian X]
(I : X.IdealSheafData) (hI : (I.support : Set X) = Set.univ) :
∃ n : ℕ, 0 < n ∧ I ^ n = ⊥ :=
exists_pos_pow_eq_bot_of_fullSupport I (SetLike.coe_injective hI)
theorem exists_pos_nilradical_pow_eq_bot [IsNoetherian X] :
∃ n : ℕ, 0 < n ∧ X.nilradical ^ n = ⊥ :=
exists_pos_pow_eq_bot_of_fullSupport X.nilradical X.support_nilradical
end
end PiExponentSeshadri.Geometry
end OAI
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