| import Mathlib.AlgebraicGeometry.IdealSheaf.Functorial |
| import Mathlib.AlgebraicGeometry.Morphisms.Separated |
| import OAI.NumberTheory.PiExponent.Ampleness.ReesPushdown |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open CategoryTheory CategoryTheory.Limits AlgebraicGeometry TopologicalSpace |
| namespace PiExponent.CompactJetIdeal |
|
|
| universe u |
| variable {U X S : Scheme.{u}} |
|
|
| def extend (I : U.IdealSheafData) (j : U ⟶ X) : X.IdealSheafData := I.map j |
|
|
| theorem restrict_extend (I : U.IdealSheafData) (j : U ⟶ X) |
| [IsOpenImmersion j] [QuasiCompact j] : (extend I j).comap j = I := by |
| apply Scheme.IdealSheafData.ext |
| funext W |
| rw [Scheme.IdealSheafData.ideal_comap_of_isOpenImmersion] |
| have hsq : IsPullback I.subschemeι (𝟙 I.subscheme) j (I.subschemeι ≫ j) := |
| IsPullback.of_vert_isIso_mono ⟨by simp⟩ |
| have h := Scheme.ker_ideal_of_isPullback_of_isOpenImmersion |
| (I.subschemeι ≫ j) I.subschemeι (𝟙 I.subscheme) j hsq W |
| simpa only [Scheme.IdealSheafData.ker_subschemeι, extend, Scheme.IdealSheafData.map] using h.symm |
|
|
| theorem section_isClosedImmersion (π : X ⟶ S) [IsSeparated π] |
| (p : S ⟶ X) (hp : p ≫ π = 𝟙 S) : IsClosedImmersion p := by |
| have : IsClosedImmersion (p ≫ π) := by rw [hp]; infer_instance |
| exact IsClosedImmersion.of_comp p π |
| |
| theorem image_support_isClosed {J : Type*} [Finite J] |
| (I : U.IdealSheafData) (j : U ⟶ X) |
| (π : X ⟶ S) [IsSeparated π] |
| (p : J → (S ⟶ U)) (hp : ∀ a, (p a ≫ j) ≫ π = 𝟙 S) |
| (hsupport : (I.support : Set U) = ⋃ a, Set.range (p a)) : |
| IsClosed (j '' (I.support : Set U)) := by |
| have heq : j '' (I.support : Set U) = ⋃ a, Set.range (p a ≫ j) := by |
| rw [hsupport, Set.image_iUnion] |
| congr 1 |
| funext a |
| rw [Scheme.Hom.comp_base, TopCat.coe_comp, Set.range_comp] |
| rw [heq] |
| apply isClosed_iUnion_of_finite |
| intro a |
| have := section_isClosedImmersion π (p a ≫ j) (hp a) |
| exact (p a ≫ j).isClosedEmbedding.isClosed_range |
|
|
| theorem support_extend {J : Type*} [Finite J] |
| (I : U.IdealSheafData) (j : U ⟶ X) [QuasiCompact j] |
| (π : X ⟶ S) [IsSeparated π] |
| (p : J → (S ⟶ U)) (hp : ∀ a, (p a ≫ j) ≫ π = 𝟙 S) |
| (hsupport : (I.support : Set U) = ⋃ a, Set.range (p a)) : |
| ((extend I j).support : Set X) = j '' (I.support : Set U) := by |
| change ((I.map j).support : Set X) = _ |
| rw [Scheme.IdealSheafData.support_map] |
| exact (image_support_isClosed I j π p hp hsupport).closure_eq |
|
|
| theorem restrict_eq_top_of_disjoint (I : X.IdealSheafData) |
| {V : Scheme.{u}} (f : V ⟶ X) |
| (havoid : ∀ v, f v ∉ I.support) : I.comap f = ⊤ := by |
| apply (Scheme.IdealSheafData.support_eq_bot_iff (I.comap f)).mp |
| rw [Scheme.IdealSheafData.support_comap] |
| ext v |
| exact ⟨fun h => (havoid v h).elim, False.elim⟩ |
|
|
| abbrev affineBlowup {R : Type u} [CommRing R] (I : Ideal R) : Scheme.{u} := |
| PiExponentSeshadri.ReesGrading.affineBlowup I |
|
|
| end PiExponent.CompactJetIdeal |
| end |
|
|
| end OAI |
|
|
