| import OAI.NumberTheory.PiExponent.Geometry.PlaceCenteredBranch |
| import OAI.NumberTheory.PiExponent.Jets.CompactLogJetIdeal |
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|
| namespace OAI |
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|
| noncomputable section |
| open CategoryTheory AlgebraicGeometry |
| namespace PiExponent.JetCenterIdentification |
| open CurveValuationCenter PlaceValuationRing CompactLogJetIdeal CompactJetPolynomial |
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|
| variable {E : Type} [Field E] [Algebra ℂ E] |
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|
| theorem centered_of_closedPoint_eq {m : ℕ} (p : NormalizedPlace ℂ E) |
| (z : Fin (m+1) → E) (c : Fin m → ℂ) |
| {X : Scheme} (j : affineSpace m ⟶ X) [IsOpenImmersion j] |
| (q : Spec (CommRingCat.of (ring p)) ⟶ X) |
| (hgeneric : Spec.map (CommRingCat.ofHom (algebraMap (ring p) E)) ≫ q = |
| Spec.map (CommRingCat.ofHom (MvPolynomial.aeval z).toRingHom) ≫ j) |
| (hclosed : q (IsLocalRing.closedPoint (ring p)) = j (centerPoint c)) : |
| CurveCenters.Centered z (Fin.cases 1 c) p := by |
| have hr : Set.range q ⊆ Set.range j := by |
| rintro _ ⟨y, rfl⟩ |
| exact ((IsLocalRing.specializes_closedPoint y).map q.continuous).mem_open |
| j.isOpenEmbedding.isOpen_range (hclosed ▸ Set.mem_range_self (centerPoint c)) |
| let g := IsOpenImmersion.lift j q hr |
| have hg : g ≫ j = q := IsOpenImmersion.lift_fac j q hr |
| let φ : coordinateRing m →+* ring p := (Spec.preimage g).hom |
| have hgφ : Spec.map (CommRingCat.ofHom φ) = g := Spec.map_preimage g |
| have hcomp : CommRingCat.ofHom φ ≫ CommRingCat.ofHom (algebraMap (ring p) E) = |
| CommRingCat.ofHom (MvPolynomial.aeval z).toRingHom := by |
| apply Spec.map_injective |
| rw [Spec.map_comp, hgφ] |
| apply (cancel_mono j).mp |
| rw [Category.assoc, hg] |
| exact hgeneric |
| have hgc : g (IsLocalRing.closedPoint (ring p)) = centerPoint c := by |
| apply j.isOpenEmbedding.injective |
| change (g ≫ j) (IsLocalRing.closedPoint (ring p)) = _ |
| rw [hg, hclosed] |
| have hcomap : PrimeSpectrum.comap φ (IsLocalRing.closedPoint (ring p)) = |
| centerPoint c := by |
| change (Spec.map (CommRingCat.ofHom φ)) (IsLocalRing.closedPoint (ring p)) = _ |
| rw [hgφ, hgc] |
| intro i |
| let P : coordinateRing m := MvPolynomial.X i - MvPolynomial.C (center c i) |
| have hmem : φ P ∈ IsLocalRing.maximalIdeal (ring p) := by |
| change P ∈ (PrimeSpectrum.comap φ (IsLocalRing.closedPoint (ring p))).asIdeal |
| rw [hcomap] |
| change P ∈ WeightedBezout.pointIdeal (center c) |
| rw [WeightedBezout.mem_pointIdeal] |
| simp [P] |
| have hpos := p.valuation.toValuation.mem_maximalIdeal_iff.mp hmem |
| have heval := congrArg |
| (fun f : CommRingCat.of (coordinateRing m) ⟶ CommRingCat.of E => f P) hcomp |
| change algebraMap (ring p) E (φ P) = MvPolynomial.aeval z P at heval |
| change 0 < p.valuation (algebraMap (ring p) E (φ P)) at hpos |
| rw [heval] at hpos |
| simpa [P, center] using hpos |
|
|
| end PiExponent.JetCenterIdentification |
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|
| end |
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| end OAI |
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