| 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 |
| |
