| 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 |
|
|
