| import OAI.NumberTheory.PiExponent.Jets.CompactLogJetIdeal |
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| namespace OAI |
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| noncomputable section |
| open scoped BigOperators |
| namespace PiExponent.JetProductLocalization |
| open CompactJetPolynomial LogarithmicContactIdeal |
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|
| variable {A : Type*} [CommRing A] [IsLocalRing A] [Algebra ℂ A] |
| [Algebra.IsIntegral ℂ (IsLocalRing.ResidueField A)] |
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| def residueAlgHom : A →ₐ[ℂ] ℂ where |
| __ := CurveLocalOrder.residueAugmentation ℂ A |
| commutes' := CurveLocalOrder.residueAugmentation_algebraMap ℂ A |
|
|
| theorem residue_ne_zero_iff_isUnit (a : A) : |
| CurveLocalOrder.residueAugmentation ℂ A a ≠ 0 ↔ IsUnit a := by |
| change (CurveLocalOrder.residueCoefficientEquiv ℂ A).symm |
| ((IsLocalRing.residue A) a) ≠ 0 ↔ IsUnit a |
| rw [map_ne_zero_iff _ (CurveLocalOrder.residueCoefficientEquiv ℂ A).symm.injective, |
| IsLocalRing.residue_ne_zero_iff_isUnit] |
|
|
| theorem residue_polynomial_aeval (a : A) (P : Polynomial ℂ) : |
| CurveLocalOrder.residueAugmentation ℂ A (Polynomial.aeval a P) = |
| P.eval (CurveLocalOrder.residueAugmentation ℂ A a) := by |
| simpa [Polynomial.coe_aeval_eq_eval, residueAlgHom] using |
| (Polynomial.aeval_algHom_apply (residueAlgHom (A := A)) a P).symm |
|
|
| theorem residue_truncatedCoordinates_succ {m : ℕ} |
| (c : Fin m → ℂ) (T : Fin m → ℕ) (y : A) (x : Fin m → A) |
| (hy : CurveLocalOrder.residueAugmentation ℂ A y = 1) (i : Fin m) : |
| CurveLocalOrder.residueAugmentation ℂ A (truncatedCoordinates c y x T i.succ) = |
| CurveLocalOrder.residueAugmentation ℂ A (x i) - c i := by |
| have hlog : CurveLocalOrder.residueAugmentation ℂ A |
| (Polynomial.aeval (y-1) (PowerSeries.trunc (T i) (PowerSeries.log ℂ))) = 0 := by |
| rw [residue_polynomial_aeval, map_sub, map_one, hy, sub_self] |
| exact logPolynomials_eval_zero T i |
| simp only [truncatedCoordinates, Fin.cases_succ, map_sub, |
| CurveLocalOrder.residueAugmentation_algebraMap, hlog, sub_zero] |
|
|
| theorem logarithmicIdeal_eq_top_of_residue_ne {m : ℕ} |
| (c : Fin m → ℂ) (T : Fin m → ℕ) (e : Fin (m+1) → ℕ) |
| (y : A) (x : Fin m → A) |
| (hne : ¬ (CurveLocalOrder.residueAugmentation ℂ A y = 1 ∧ |
| ∀ i, CurveLocalOrder.residueAugmentation ℂ A (x i) = c i)) : |
| logarithmicIdeal c y x T e = ⊤ := by |
| classical |
| have hunit : ∃ i : Fin (m+1), IsUnit (truncatedCoordinates c y x T i) := by |
| by_cases hy : CurveLocalOrder.residueAugmentation ℂ A y = 1 |
| · have hx : ¬ ∀ i, CurveLocalOrder.residueAugmentation ℂ A (x i) = c i := |
| fun hx => hne ⟨hy, hx⟩ |
| obtain ⟨i, hi⟩ := not_forall.mp hx |
| refine ⟨i.succ, (residue_ne_zero_iff_isUnit _).mp ?_⟩ |
| rw [residue_truncatedCoordinates_succ c T y x hy i] |
| exact sub_ne_zero.mpr hi |
| · refine ⟨0, (residue_ne_zero_iff_isUnit _).mp ?_⟩ |
| change CurveLocalOrder.residueAugmentation ℂ A (y-1) ≠ 0 |
| rw [map_sub, map_one] |
| exact sub_ne_zero.mpr hy |
| obtain ⟨i, hi⟩ := hunit |
| apply Ideal.eq_top_of_isUnit_mem _ |
| (show truncatedCoordinates c y x T i ^ e i ∈ logarithmicIdeal c y x T e from |
| Ideal.subset_span (Set.mem_range_self i)) |
| exact hi.pow (e i) |
|
|
| theorem map_powerIdeal_eq_top_of_ne {J : Type*} {m : ℕ} |
| (c : J → Fin m → ℂ) (hc : Function.Injective c) |
| (T : Fin m → ℕ) (e : Fin (m+1) → ℕ) (y : A) (x : Fin m → A) |
| (j k : J) (hkj : k ≠ j) |
| (hy : CurveLocalOrder.residueAugmentation ℂ A y = 1) |
| (hx : ∀ i, CurveLocalOrder.residueAugmentation ℂ A (x i) = c j i) : |
| (powerIdeal (c k) (logPolynomials T) e).map |
| (MvPolynomial.aeval (Fin.cases y x)).toRingHom = ⊤ := by |
| rw [map_powerIdeal_aeval] |
| apply logarithmicIdeal_eq_top_of_residue_ne |
| rw [hy] |
| rintro ⟨_, hk⟩ |
| apply hkj |
| apply hc |
| funext i |
| exact (hk i).symm.trans (hx i) |
|
|
| omit [IsLocalRing A] [Algebra.IsIntegral ℂ (IsLocalRing.ResidueField A)] in |
| theorem map_polynomialIdeal {J : Type*} [Fintype J] {m : ℕ} |
| (c : J → Fin m → ℂ) (T : Fin m → ℕ) (e : Fin (m+1) → ℕ) |
| (y : A) (x : Fin m → A) : |
| (CompactLogJetIdeal.polynomialIdeal c T e).map |
| (MvPolynomial.aeval (Fin.cases y x)).toRingHom = |
| ∏ j, logarithmicIdeal (c j) y x T e := by |
| unfold CompactLogJetIdeal.polynomialIdeal |
| change Ideal.mapHom (MvPolynomial.aeval (Fin.cases y x)).toRingHom _ = _ |
| rw [map_prod] |
| apply Finset.prod_congr rfl |
| intro j _ |
| exact map_powerIdeal_aeval (c j) T e y x |
|
|
| theorem map_polynomialIdeal_eq_selected {J : Type*} [Fintype J] {m : ℕ} |
| (c : J → Fin m → ℂ) (hc : Function.Injective c) |
| (T : Fin m → ℕ) (e : Fin (m+1) → ℕ) (y : A) (x : Fin m → A) (j : J) |
| (hy : CurveLocalOrder.residueAugmentation ℂ A y = 1) |
| (hx : ∀ i, CurveLocalOrder.residueAugmentation ℂ A (x i) = c j i) : |
| (CompactLogJetIdeal.polynomialIdeal c T e).map |
| (MvPolynomial.aeval (Fin.cases y x)).toRingHom = logarithmicIdeal (c j) y x T e := by |
| classical |
| rw [map_polynomialIdeal] |
| apply Finset.prod_eq_single j |
| · intro k _ hkj |
| rw [Ideal.one_eq_top] |
| rw [← map_powerIdeal_aeval] |
| exact map_powerIdeal_eq_top_of_ne c hc T e y x j k hkj hy hx |
| · simp |
|
|
| theorem map_polynomialIdeal_eq_top_outside {J : Type*} [Fintype J] {m : ℕ} |
| (c : J → Fin m → ℂ) (T : Fin m → ℕ) (e : Fin (m+1) → ℕ) |
| (y : A) (x : Fin m → A) |
| (hout : ¬ ∃ j, CurveLocalOrder.residueAugmentation ℂ A y = 1 ∧ |
| ∀ i, CurveLocalOrder.residueAugmentation ℂ A (x i) = c j i) : |
| (CompactLogJetIdeal.polynomialIdeal c T e).map |
| (MvPolynomial.aeval (Fin.cases y x)).toRingHom = ⊤ := by |
| rw [map_polynomialIdeal] |
| rw [← Ideal.one_eq_top] |
| apply Finset.prod_eq_one |
| intro j _ |
| rw [Ideal.one_eq_top] |
| exact logarithmicIdeal_eq_top_of_residue_ne (c j) T e y x (fun h => hout ⟨j, h⟩) |
|
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| end PiExponent.JetProductLocalization |
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| end |
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| end OAI |
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