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b797a5b | 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 58 59 60 | import OAI.NumberTheory.PiExponent.Geometry.CurveImageAffineCoordinates
namespace OAI
noncomputable section
namespace PiExponent.CurveImageAffineCoordinates
open AlgebraicGeometry CategoryTheory TopologicalSpace
variable {C X : Scheme.{0}} {ι : Type}
def affinePart (i : C ⟶ X) [IsClosedImmersion i]
(j : Spec (.of (MvPolynomial ι ℂ)) ⟶ X) [IsOpenImmersion j] : C.affineOpens :=
⟨i ⁻¹ᵁ j.opensRange, (isAffineOpen_opensRange j).preimage i⟩
def affinePresentation (i : C ⟶ X) [IsClosedImmersion i]
(j : Spec (.of (MvPolynomial ι ℂ)) ⟶ X) [IsOpenImmersion j] :
Spec Γ(C,(affinePart i j).1) ⟶ Spec (.of (MvPolynomial ι ℂ)) :=
(affinePart i j).2.isoSpec.inv ≫ (i ∣_ j.opensRange) ≫ j.isoOpensRange.inv
instance affinePresentation_isClosedImmersion (i : C ⟶ X) [IsClosedImmersion i]
(j : Spec (.of (MvPolynomial ι ℂ)) ⟶ X) [IsOpenImmersion j] :
IsClosedImmersion (affinePresentation i j) := by
unfold affinePresentation
have hcomp : IsClosedImmersion ((i ∣_ j.opensRange) ≫ j.isoOpensRange.inv) :=
IsClosedImmersion.comp _ _
exact @IsClosedImmersion.comp _ _ _ _ _ inferInstance hcomp
@[reassoc] theorem affinePresentation_comp (i : C ⟶ X) [IsClosedImmersion i]
(j : Spec (.of (MvPolynomial ι ℂ)) ⟶ X) [IsOpenImmersion j] :
affinePresentation i j ≫ j = (affinePart i j).2.fromSpec ≫ i := by
unfold affinePresentation IsAffineOpen.fromSpec
exact (Category.assoc _ _ _).trans
((congrArg (fun k => (affinePart i j).2.isoSpec.inv ≫ k)
((Category.assoc _ _ _).trans
((congrArg (fun k => (i ∣_ j.opensRange) ≫ k)
(Scheme.Hom.isoOpensRange_inv_comp j)).trans
(morphismRestrict_ι i j.opensRange)))).trans
(Category.assoc _ _ _).symm)
theorem affinePresentation_over (p : X ⟶ Spec (.of ℂ))
(i : C ⟶ X) [IsClosedImmersion i]
(j : Spec (.of (MvPolynomial ι ℂ)) ⟶ X) [IsOpenImmersion j]
(hj : j ≫ p = Spec.map (CommRingCat.ofHom
(MvPolynomial.C : ℂ →+* MvPolynomial ι ℂ))) :
affinePresentation i j ≫ Spec.map (CommRingCat.ofHom
(MvPolynomial.C : ℂ →+* MvPolynomial ι ℂ)) =
(affinePart i j).2.fromSpec ≫ i ≫ p := by
rw [← hj, ← Category.assoc, affinePresentation_comp, Category.assoc]
theorem affinePart_nonempty (i : C ⟶ X) [IsClosedImmersion i]
(j : Spec (.of (MvPolynomial ι ℂ)) ⟶ X) [IsOpenImmersion j]
(h : ∃ c : C, i c ∈ j.opensRange) : Nonempty (affinePart i j).1 := by
obtain ⟨c,hc⟩ := h
exact ⟨⟨c,hc⟩⟩
end PiExponent.CurveImageAffineCoordinates
end
end OAI
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