| import Mathlib |
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| namespace OAI |
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| section |
| noncomputable section |
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| namespace MaximalSeshadri.ProjectiveBertini |
| noncomputable section |
| open AlgebraicGeometry CategoryTheory TopologicalSpace |
| universe u |
| variable {K : Type u} [Field K] {X : Scheme.{u}} |
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| def openScalars (g : X ⟶ Spec (CommRingCat.of K)) (U : X.Opens) : K →+* Γ(X, U) := |
| (g.appLE ⊤ U (by simp)).hom.comp (Scheme.ΓSpecIso (CommRingCat.of K)).inv.hom |
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| lemma openScalars_restrict (g : X ⟶ Spec (CommRingCat.of K)) {U V : X.Opens} |
| (e : U ≤ V) : |
| (X.presheaf.map (homOfLE e).op).hom.comp (openScalars g V) = openScalars g U := by |
| exact congrArg (fun morphism : Γ(Spec (CommRingCat.of K), ⊤) ⟶ Γ(X, U) => |
| morphism.hom.comp (Scheme.ΓSpecIso (CommRingCat.of K)).inv.hom) |
| (Scheme.Hom.appLE_map g (show V ≤ g ⁻¹ᵁ ⊤ from by simp) (homOfLE e).op) |
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| lemma spec_openScalars (g : X ⟶ Spec (CommRingCat.of K)) (U : X.affineOpens) : |
| Spec.map (CommRingCat.ofHom (openScalars g U.1)) = U.2.fromSpec ≫ g := by |
| rw [openScalars, CommRingCat.ofHom_comp, Spec.map_comp, CommRingCat.ofHom_hom, |
| CommRingCat.ofHom_hom] |
| rw [← Scheme.isoSpec_Spec_inv, ← IsAffineOpen.fromSpec_top] |
| exact IsAffineOpen.SpecMap_appLE_fromSpec g (isAffineOpen_top _) U.2 _ |
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| theorem exists_standard_smooth_affine_inside |
| (g : X ⟶ Spec (CommRingCat.of K)) (n : ℕ) [SmoothOfRelativeDimension n g] |
| (W : X.Opens) (x : X) (hx : x ∈ W) : |
| ∃ U : X.affineOpens, x ∈ U.1 ∧ U.1 ≤ W ∧ |
| RingHom.IsStandardSmoothOfRelativeDimension n (openScalars g U.1) := by |
| obtain ⟨B, hB, V, hV, hxV, e, hg⟩ := |
| SmoothOfRelativeDimension.exists_isStandardSmoothOfRelativeDimension |
| (n := n) (f := g) x |
| have hBtop : B = ⊤ := by |
| apply top_unique |
| intro y _ |
| have hy : y = g x := Subsingleton.elim _ _ |
| rw [hy] |
| exact e hxV |
| subst B |
| let e₀ : K ≃+* Γ(Spec (CommRingCat.of K), ⊤) := |
| RingEquiv.ofBijective (Scheme.ΓSpecIso (CommRingCat.of K)).inv.hom |
| (ConcreteCategory.bijective_of_isIso (Scheme.ΓSpecIso (CommRingCat.of K)).inv) |
| have hstd : RingHom.IsStandardSmoothOfRelativeDimension n (openScalars g V) := by |
| convert hg.comp (RingHom.IsStandardSmoothOfRelativeDimension.equiv e₀) using 1 |
| rfl |
| obtain ⟨f, hfW, hxf⟩ := hV.exists_basicOpen_le ⟨x, hx⟩ hxV |
| refine ⟨⟨X.basicOpen f, hV.basicOpen f⟩, hxf, hfW, ?_⟩ |
| let : IsLocalization.Away f Γ(X, X.basicOpen f) := hV.isLocalization_basicOpen f |
| have hs := (RingHom.isStandardSmoothOfRelativeDimension_stableUnderCompositionWithLocalizationAway n).right |
| (Γ(X, X.basicOpen f)) f (openScalars g V) hstd |
| convert hs using 1 |
| exact (openScalars_restrict g (X.basicOpen_le f)).symm |
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| end |
| end MaximalSeshadri.ProjectiveBertini |
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| end |
| end |
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| end OAI |
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