File size: 4,118 Bytes
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 | import OAI.NumberTheory.PiExponent.Cohomology.EulerSupportDimension
import OAI.NumberTheory.PiExponent.Geometry.SectionZeroStalk
import OAI.NumberTheory.PiExponent.LocalAlgebra.LocalIntersectionMinimalSum
namespace OAI
namespace PiExponent.CurveCycle
noncomputable section
open AlgebraicGeometry CategoryTheory TopologicalSpace
open PiExponentSeshadri.Geometry PiExponentSeshadri.Frames
open PiExponentJets.W28.LocalIntersection
open scoped nonZeroDivisors
theorem stalk_krullDimLE_one (X : Scheme.{0})
(hd : topologicalKrullDim X ≤ 1) (x : X) :
Ring.KrullDimLE 1 (X.presheaf.stalk x) := by
apply Ring.krullDimLE_iff.mpr
rw [← PrimeSpectrum.topologicalKrullDim_eq_ringKrullDim]
exact (X.fromSpecStalk x).isEmbedding.isInducing.topologicalKrullDim_le.trans hd
theorem sectionGerm_nonZeroDivisor {X : Scheme.{0}}
(L : LineBundle X) (s : GlobalSections X L.sheaf) [Mono s]
(U : X.affineOpens) (e : L.sheaf.restrict U.1.ι ≅ O U.1.toScheme)
(x : X) (hx : x ∈ U.1) :
SectionZeroStalk.sectionGerm L s U e x hx ∈ nonZeroDivisors (X.presheaf.stalk x) := by
let := X.presheaf.algebra_section_stalk (⟨x,hx⟩ : U.1)
let := U.2.isLocalization_stalk (⟨x,hx⟩ : U.1)
exact IsLocalization.nonZeroDivisors_le_comap
((U.2.primeIdealOf ⟨x,hx⟩).asIdeal.primeCompl) (X.presheaf.stalk x)
(NumericalAmpleness.section_affine_coefficient_nonZeroDivisor L s U e)
theorem zero_stalk_length_eq_minimalPrimeCutSum {X : Scheme.{0}}
[IsLocallyNoetherian X] (hd : topologicalKrullDim X ≤ 1)
(L : LineBundle X) (s : GlobalSections X L.sheaf) [Mono s]
(U : X.affineOpens) (e : L.sheaf.restrict U.1.ι ≅ O U.1.toScheme)
(y : (SectionZeroIdeal.zeroIdeal L s).subscheme)
(hy : (SectionZeroIdeal.zeroIdeal L s).subschemeι y ∈ U.1) :
Module.length ((SectionZeroIdeal.zeroIdeal L s).subscheme.presheaf.stalk y)
((SectionZeroIdeal.zeroIdeal L s).subscheme.presheaf.stalk y) =
minimalPrimeCutSum (⊥ : Ideal
(X.presheaf.stalk ((SectionZeroIdeal.zeroIdeal L s).subschemeι y)))
(SectionZeroStalk.sectionGerm L s U e _ hy) := by
let x := (SectionZeroIdeal.zeroIdeal L s).subschemeι y
let A := X.presheaf.stalk x
let c : A := SectionZeroStalk.sectionGerm L s U e x hy
let := stalk_krullDimLE_one X hd x
have hlen := SectionZeroStalk.intrinsic_length_eq_of_ringEquiv
(SectionZeroStalk.zeroStalkQuotientEquiv L s U e y hy)
have hsc : Module.length A (A ⧸ Ideal.span {c}) =
Module.length (A ⧸ Ideal.span {c}) (A ⧸ Ideal.span {c}) :=
Module.length_eq_of_surjective (M := A ⧸ Ideal.span {c}) Ideal.Quotient.mk_surjective
exact hlen.symm.trans (hsc.symm.trans
(regular_ring_onecut_length c (sectionGerm_nonZeroDivisor L s U e x hy)))
theorem zero_stalk_length_ne_top {X : Scheme.{0}} [IsLocallyNoetherian X]
(hd : topologicalKrullDim X ≤ 1)
(L : LineBundle X) (s : GlobalSections X L.sheaf) [Mono s]
(y : (SectionZeroIdeal.zeroIdeal L s).subscheme) :
Module.length ((SectionZeroIdeal.zeroIdeal L s).subscheme.presheaf.stalk y)
((SectionZeroIdeal.zeroIdeal L s).subscheme.presheaf.stalk y) ≠ ⊤ := by
let x := (SectionZeroIdeal.zeroIdeal L s).subschemeι y
obtain ⟨V,hxV,⟨eV⟩⟩ := L.locallyRankOne x
obtain ⟨W,hW,hxW,hWV⟩ := exists_isAffineOpen_mem_and_subset hxV
let U : X.affineOpens := ⟨W,hW⟩
let e := restrictOpenFrame hWV eV
let A := X.presheaf.stalk x
let c : A := SectionZeroStalk.sectionGerm L s U e x hxW
let := stalk_krullDimLE_one X hd x
have hfinite : IsFiniteLength A (A ⧸ Ideal.span {c}) :=
isFiniteLength_quotient_span_singleton A (sectionGerm_nonZeroDivisor L s U e x hxW)
have hsc : Module.length A (A ⧸ Ideal.span {c}) =
Module.length (A ⧸ Ideal.span {c}) (A ⧸ Ideal.span {c}) :=
Module.length_eq_of_surjective (M := A ⧸ Ideal.span {c}) Ideal.Quotient.mk_surjective
have hlen := SectionZeroStalk.intrinsic_length_eq_of_ringEquiv
(SectionZeroStalk.zeroStalkQuotientEquiv L s U e y hxW)
rw [← hlen,← hsc]
exact Module.length_ne_top_iff.mpr hfinite
end
end PiExponent.CurveCycle
end OAI
|