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12dee50 | 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 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 | import Mathlib
import OAI.AlgebraicGeometry.Seshadri.Analytic.EtaleJetCharts
namespace OAI
section
namespace MaximalSeshadri.Geometry
noncomputable section
open AlgebraicGeometry CategoryTheory TopologicalSpace
open MaximalSeshadri.AlgebraicJets MaximalSeshadri.AnalyticCoordinates
lemma formalTrunc_zero_iff_order (f : MvPowerSeries (Fin 2) ℂ) (n : ℕ) :
formalTrunc n f = 0 ↔ (n : ℕ∞) ≤ f.order := by
change Ideal.Quotient.mk _ (MvPowerSeries.truncTotal n f) = 0 ↔ _
rw [Ideal.Quotient.eq_zero_iff_mem,MvPolynomial.mem_pow_idealOfVars_iff']
constructor
· intro h
apply MvPowerSeries.nat_le_order
intro d hd
simpa only [MvPowerSeries.coeff_truncTotal f hd] using h d hd
· intro h d hd
rw [MvPowerSeries.coeff_truncTotal f hd]
apply MvPowerSeries.coeff_of_lt_order
exact lt_of_lt_of_le (by exact_mod_cast hd) h
lemma taylor_mem_power_iff_order {R : Type} [CommRing R] [Algebra ℂ R]
(ρ : R →ₐ[ℂ] ℂ) (τ : R →ₐ[ℂ] MvPowerSeries (Fin 2) ℂ)
(hτ : ∀ n : ℕ, 0 < n → RingHom.ker ((formalTrunc n).comp τ) = (RingHom.ker ρ)^n)
(a : R) (n : ℕ) : a ∈ (RingHom.ker ρ)^n ↔ (n : ℕ∞) ≤ (τ a).order := by
by_cases hn : n = 0
· subst n; simp
· rw [← hτ n (Nat.pos_of_ne_zero hn)]
change formalTrunc n (τ a) = 0 ↔ _
exact formalTrunc_zero_iff_order _ _
lemma taylor_cancel_power {R : Type} [CommRing R] [Algebra ℂ R]
(ρ : R →ₐ[ℂ] ℂ) (τ : R →ₐ[ℂ] MvPowerSeries (Fin 2) ℂ)
(hτ : ∀ n : ℕ, 0 < n → RingHom.ker ((formalTrunc n).comp τ) = (RingHom.ker ρ)^n)
(a b : R) (n m : ℕ) (ha : a ∉ (RingHom.ker ρ)^(m+1))
(hab : a*b ∈ (RingHom.ker ρ)^n) : b ∈ (RingHom.ker ρ)^(n-m) := by
by_contra hb
rw [taylor_mem_power_iff_order ρ τ hτ] at ha hb hab
have ha' := lt_of_not_ge ha
have hb' := lt_of_not_ge hb
have hat : (τ a).order ≠ ⊤ := ne_top_of_lt ha'
have hbt : (τ b).order ≠ ⊤ := ne_top_of_lt hb'
rw [← ENat.natCast_toNat hat] at ha'
rw [← ENat.natCast_toNat hbt] at hb'
have haN : (τ a).order.toNat < m+1 := by exact_mod_cast ha'
have hbN : (τ b).order.toNat < n-m := by exact_mod_cast hb'
rw [map_mul,MvPowerSeries.order_mul,← ENat.natCast_toNat hat,
← ENat.natCast_toNat hbt,← Nat.cast_add] at hab
have habN : n ≤ (τ a).order.toNat+(τ b).order.toNat := by exact_mod_cast hab
omega
theorem standard_smooth_cancel_power {R : Type} [CommRing R] [IsDomain R] [Algebra ℂ R]
[Algebra.IsStandardSmoothOfRelativeDimension 2 ℂ R]
(ρ : R →ₐ[ℂ] ℂ) (a b : R) (n m : ℕ)
(ha : a ∉ (RingHom.ker ρ)^(m+1)) (hab : a*b ∈ (RingHom.ker ρ)^n) :
b ∈ (RingHom.ker ρ)^(n-m) := by
classical
obtain ⟨ι,σ,hσ,hι,P,hP⟩ := Algebra.IsStandardSmoothOfRelativeDimension.out (n := 2) (R := ℂ) (S := R)
let := hσ
let := hι
let := Fintype.ofFinite σ
let := Fintype.ofFinite ι
let e : ((Set.range P.map)ᶜ : Set ι) ≃ Fin 2 := Fintype.equivFinOfCardEq (by
rw [← hP]
simp only [Algebra.Presentation.dimension, Nat.card_eq_fintype_card,
Fintype.card_compl_set, Set.card_range_of_injective P.map_inj])
obtain ⟨q,hq,-,-,hjets,-⟩ := presentation_analytic_realization P e ρ
exact taylor_cancel_power ρ (analyticTaylor q hq) hjets a b n m ha hab
end
end MaximalSeshadri.Geometry
end
end OAI
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