| import Mathlib |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace CampanaPeternellSix.RelativeTriviality |
| open CategoryTheory AlgebraicGeometry Limits |
| universe u |
| attribute [local instance] MvPolynomial.gradedAlgebra |
| variable (R : Type u) [CommRing R] |
| abbrev grading := MvPolynomial.homogeneousSubmodule (Fin 2) R |
| def line : Scheme := Proj (grading R) |
| def constants : R →+* grading R 0 where |
| toFun r := ⟨MvPolynomial.C r, MvPolynomial.isHomogeneous_C _ r⟩ |
| map_one' := Subtype.ext (map_one _) |
| map_mul' a b := Subtype.ext (map_mul _ a b) |
| map_zero' := Subtype.ext (map_zero _) |
| map_add' a b := Subtype.ext (map_add _ a b) |
| def toBase : line R ⟶ Spec (.of R) := |
| AlgebraicGeometry.Proj.toSpecZero (grading R) ≫ Spec.map (CommRingCat.ofHom (constants R)) |
|
|
| def fiberProjection (p : PrimeSpectrum R) : |
| pullback (toBase R) (Spec.map (CommRingCat.ofHom (algebraMap R p.asIdeal.ResidueField))) ⟶ line R := |
| pullback.fst _ _ |
|
|
| def awayProjection (r : R) : |
| pullback (toBase R) (Spec.map (CommRingCat.ofHom (algebraMap R (Localization.Away r)))) ⟶ line R := |
| pullback.fst _ _ |
| |
| instance constants_isIso : IsIso (CommRingCat.ofHom (constants R)) := by |
| rw [ConcreteCategory.isIso_iff_bijective] |
| constructor |
| · intro r s h |
| exact MvPolynomial.C_injective (Fin 2) R (congrArg Subtype.val h) |
| · intro p |
| refine ⟨p.val.coeff 0, Subtype.ext ?_⟩ |
| exact (MvPolynomial.totalDegree_eq_zero_iff_eq_C.mp |
| ((MvPolynomial.totalDegree_zero_iff_isHomogeneous _).mpr p.property)).symm |
| instance line_toBase_universallyClosed : UniversallyClosed (toBase R) := by |
| have : Algebra.FiniteType (grading R 0) (MvPolynomial (Fin 2) R) := by |
| apply RingHom.finiteType_algebraMap.mp |
| apply RingHom.FiniteType.of_comp_finiteType (f := constants R) |
| change (algebraMap R (MvPolynomial (Fin 2) R)).FiniteType |
| exact RingHom.finiteType_algebraMap.mpr inferInstance |
| have hproj : UniversallyClosed (AlgebraicGeometry.Proj.toSpecZero (grading R)) := inferInstance |
| have : IsIso (Spec.map (CommRingCat.ofHom (constants R))) := inferInstance |
| have : IsClosedImmersion (Spec.map (CommRingCat.ofHom (constants R))) := inferInstance |
| have hspec : UniversallyClosed (Spec.map (CommRingCat.ofHom (constants R))) := inferInstance |
| unfold toBase |
| exact @AlgebraicGeometry.universallyClosedTypeComp _ _ _ _ _ hproj hspec |
| end CampanaPeternellSix.RelativeTriviality |
| |
| end |
| |
| end OAI |
| |
