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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 61 62 63 64 65 | import OAI.NumberTheory.PiExponent.Geometry.PlaceCenteredBranch
import OAI.NumberTheory.PiExponent.Jets.CompactLogJetIdeal
namespace OAI
noncomputable section
open CategoryTheory AlgebraicGeometry
namespace PiExponent.JetCenterIdentification
open CurveValuationCenter PlaceValuationRing CompactLogJetIdeal CompactJetPolynomial
variable {E : Type} [Field E] [Algebra ℂ E]
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
end
end OAI
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