| import OAI.AlgebraicGeometry.SurfaceCones.CoherentTensor |
| import OAI.AlgebraicGeometry.SurfaceCones.ReesPunctureTopology |
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|
| namespace OAI |
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|
| noncomputable section |
| namespace ExplicitCone |
| open AlgebraicGeometry CategoryTheory |
|
|
| /-- The genuine structure map of the distinguished affine chart of the |
| literal eighteen-section ordinary blowup. -/ |
| abbrev completedChartBaseMap : CommRingCat.of completedRing ⟶ |
| CommRingCat.of completedBlowupChart := |
| CommRingCat.ofHom (algebraMap completedRing completedBlowupChart) |
|
|
| lemma completedChartBaseMap_injective : Function.Injective completedChartBaseMap := by |
| intro a b hab |
| apply completedAffineSource_injective |
| calc |
| completedAffineSource a = completedBlowupChartEquiv |
| (algebraMap completedRing completedBlowupChart a) := |
| (completedBlowupChartEquiv.commutes a).symm |
| _ = completedBlowupChartEquiv (algebraMap completedRing completedBlowupChart b) := |
| congrArg completedBlowupChartEquiv hab |
| _ = completedAffineSource b := completedBlowupChartEquiv.commutes b |
|
|
| instance completedBlowupExceptional_domain : |
| IsDomain (completedBlowupChart ⧸ Ideal.span {completedBlowupParameter}) := |
| completedBlowupExceptionalEquiv.injective.isDomain completedBlowupExceptionalEquiv.toRingHom |
|
|
| instance completedBlowupExceptional_prime : |
| (Ideal.span ({completedBlowupParameter} : Set completedBlowupChart)).IsPrime := |
| (Ideal.Quotient.isDomain_iff_prime _).mp completedBlowupExceptional_domain |
|
|
| lemma completedBlowupParameter_regular : |
| IsSMulRegular completedBlowupChart completedBlowupParameter := by |
| intro x y hxy |
| exact completedBlowupChart_domain.mul_left_cancel_of_ne_zero |
| completedBlowupParameter_ne_zero hxy |
|
|
| /-- The exceptional-chart module is the restriction of the double dual of the pullback module to a prime Cartier divisor. -/ |
| abbrev completedChartRestrictionSections (M : ModuleCat.{0} completedRing) := |
| let q : CommRingCat.of completedBlowupChart ⟶ |
| CommRingCat.of (completedBlowupChart ⧸ Ideal.span {completedBlowupParameter}) := |
| CommRingCat.ofHom (Ideal.Quotient.mk _) |
| (moduleSpecΓFunctor (R := CommRingCat.of |
| (completedBlowupChart ⧸ Ideal.span {completedBlowupParameter}))).obj |
| ((Scheme.Modules.pullback (Spec.map q)).obj |
| (CoherentModelActual.E (Spec.map completedChartBaseMap) M)) |
|
|
| /-- Torsion freedom is proved for the concrete exceptional module, not |
| assumed as input to a numerical argument. There is no grading on M. -/ |
| theorem completedChartRestriction_torsionFree (M : ModuleCat.{0} completedRing) |
| [Module.Finite completedRing M] : |
| Module.IsTorsionFree (completedBlowupChart ⧸ Ideal.span {completedBlowupParameter}) |
| (completedChartRestrictionSections M) := |
| CoherentModelActual.affineE_cartier_torsionFree completedChartBaseMap M |
| completedBlowupParameter completedBlowupParameter_regular |
|
|
| /-- A nonzero finite torsion-free module gives positive rank on the exceptional affine section chart. -/ |
| theorem completedChartRestriction_rank_pos (M : ModuleCat.{0} completedRing) |
| [Module.Finite completedRing M] [Module.IsTorsionFree completedRing M] [Nontrivial M] : |
| 0 < Module.finrank (completedBlowupChart ⧸ Ideal.span {completedBlowupParameter}) |
| (completedChartRestrictionSections M) := by |
| let := actualCompletion_noetherian |
| exact CoherentModelActual.affineE_cartier_rank_pos completedChartBaseMap M |
| completedChartBaseMap_injective completedBlowupParameter |
| completedBlowupParameter_regular |
|
|
| end ExplicitCone |
|
|
|
|
| namespace ExplicitCone |
| open AlgebraicGeometry CategoryTheory |
|
|
| /-- The literal double-dual extension for an arbitrary module on the |
| completed cone, on the genuine proper Rees modification constructed above. |
| This is not an assumed object with prescribed numerical invariants. -/ |
| abbrev completedModelE (M : ModuleCat completedRing) : completedBlowup.Modules := |
| CoherentModelActual.E completedBlowupMap M |
|
|
| /-- Coherence for EVERY finite module of the completed ring: no grading, |
| algebra structure, or freeness premise is imposed on M. -/ |
| theorem completedModelE_coherent (M : ModuleCat completedRing) |
| [Module.Finite completedRing M] : (completedModelE M).IsFinitePresentation := by |
| let := actualCompletion_noetherian |
| exact CoherentModelActual.coherent_E completedBlowupMap M |
|
|
| /-- Torsion freeness of the literal extension, proved on all open-section |
| modules of the fixed integral model. No source regularity is assumed. -/ |
| theorem completedModelE_torsionFree (M : ModuleCat completedRing) |
| (U : completedBlowup.Opens) : |
| @Module.IsTorsionFree Γ(completedBlowup, U) |
| ((completedModelE M).val.obj (Opposite.op U)) _ _ |
| ((completedModelE M).val.obj (Opposite.op U)).isModule := |
| CoherentDual.dual_sections_torsionFree _ U |
|
|
| /-- The first dual in the same construction is also genuinely coherent. -/ |
| theorem completedModel_firstDual_coherent (M : ModuleCat completedRing) |
| [Module.Finite completedRing M] : |
| (CoherentDual.homSheaf completedBlowup.sheaf |
| ((Scheme.Modules.pullback completedBlowupMap).obj (tilde M)) |
| (SheafOfModules.unit _)).IsFinitePresentation := by |
| let := actualCompletion_noetherian |
| obtain ⟨P, hP⟩ := CoherentPullback.exists_presentation_pullback_tilde completedBlowupMap M |
| let := hP |
| have : ((Scheme.Modules.pullback completedBlowupMap).obj (tilde M)).IsFinitePresentation := |
| CoherentLocality.finitePresentation_of_presentation P |
| exact CoherentDual.coherent_dual _ |
|
|
| end ExplicitCone |
|
|
|
|
| namespace ExplicitCone |
| open AlgebraicGeometry CategoryTheory Scheme.Modules |
|
|
| /-- The literal prime Cartier divisor in the distinguished actual blowup |
| chart, embedded into the GLOBAL proper model. -/ |
| abbrev completedExceptionalChartMap : |
| Spec (.of (completedBlowupChart ⧸ Ideal.span {completedBlowupParameter})) ⟶ |
| completedBlowup := |
| Spec.map (CommRingCat.ofHom (Ideal.Quotient.mk (Ideal.span {completedBlowupParameter}))) ≫ |
| completedChartInclusion |
|
|
| /-- Sections of the restriction of the GLOBALLY defined extension E. -/ |
| abbrev completedGlobalRestrictionSections (M : ModuleCat.{0} completedRing) := |
| (moduleSpecΓFunctor (R := CommRingCat.of |
| (completedBlowupChart ⧸ Ideal.span {completedBlowupParameter}))).obj |
| ((pullback completedExceptionalChartMap).obj (completedModelE M)) |
|
|
| /-- Compatibility with the actual global model; this closes the local/global |
| identity left separate in the preceding positive-rank chart calculation. -/ |
| def completedGlobalRestrictionIso (M : ModuleCat.{0} completedRing) : |
| completedGlobalRestrictionSections M ≅ completedChartRestrictionSections M := by |
| let q : CommRingCat.of completedBlowupChart ⟶ |
| CommRingCat.of (completedBlowupChart ⧸ Ideal.span {completedBlowupParameter}) := |
| CommRingCat.ofHom (Ideal.Quotient.mk _) |
| let e := CoherentModelActual.openPullbackEIsoOfComp completedChartInclusion |
| completedBlowupMap (Spec.map completedChartBaseMap) completedChartInclusion_map M |
| exact (moduleSpecΓFunctor (R := CommRingCat.of |
| (completedBlowupChart ⧸ Ideal.span {completedBlowupParameter}))).mapIso |
| (((pullbackComp (Spec.map q) completedChartInclusion).app (completedModelE M)).symm ≪≫ |
| (pullback (Spec.map q)).mapIso e) |
|
|
| /-- The actual global extension restricts to a torsion-free module along |
| this whole exceptional chart, not merely its generic point. -/ |
| theorem completedGlobalRestriction_torsionFree (M : ModuleCat.{0} completedRing) |
| [Module.Finite completedRing M] : |
| Module.IsTorsionFree (completedBlowupChart ⧸ Ideal.span {completedBlowupParameter}) |
| (completedGlobalRestrictionSections M) := by |
| let e := (completedGlobalRestrictionIso M).toLinearEquiv |
| let := completedChartRestriction_torsionFree M |
| exact e.injective.moduleIsTorsionFree e e.map_smul |
|
|
| /-- The restriction of the global double-dual extension has positive rank for every nonzero finite torsion-free module. -/ |
| theorem completedGlobalRestriction_rank_pos (M : ModuleCat.{0} completedRing) |
| [Module.Finite completedRing M] [Module.IsTorsionFree completedRing M] [Nontrivial M] : |
| 0 < Module.finrank (completedBlowupChart ⧸ Ideal.span {completedBlowupParameter}) |
| (completedGlobalRestrictionSections M) := by |
| rw [(completedGlobalRestrictionIso M).toLinearEquiv.finrank_eq] |
| exact completedChartRestriction_rank_pos M |
|
|
| end ExplicitCone |
|
|
| end |
|
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| end OAI |
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|
