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7f5fab8 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 | 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
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