| import Mathlib |
| import OAI.AlgebraicGeometry.Seshadri.Analytic.EtaleJetCharts |
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| namespace OAI |
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| section |
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| namespace MaximalSeshadri.Geometry |
| noncomputable section |
| open AlgebraicGeometry CategoryTheory TopologicalSpace |
| open MaximalSeshadri.AlgebraicJets MaximalSeshadri.AnalyticCoordinates |
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| 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 |
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| 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 _ _ |
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| 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 |
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| 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 |
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| end |
| end MaximalSeshadri.Geometry |
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| end |
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| end OAI |
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