| import Mathlib |
| import OAI.Algebra.FrobeniusObstruction.Obstruction |
| import OAI.Algebra.AlgebraicObstruction.LocalOrigin |
| import OAI.Algebra.AlgebraicObstruction.TaylorTarget |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped BigOperators |
|
|
| namespace BoundaryOnly.FormalObstruction.FormalCorrection.AffineEtaleChart |
| open MvPowerSeries |
| open BoundaryOnly.FormalObstruction.AlgebraicReplacement |
| variable {K α : Type} [Field K] [Finite α] |
|
|
| omit [Finite α] in |
| lemma origin_eq_variables : origin (K := K) (α := α) = MvPolynomial.idealOfVars α K := |
| (TaylorTarget.variables_ker K α).symm |
|
|
| omit [Finite α] in |
| lemma maximalIdeal_eq_map_variables (E : AffineEtaleChart K α) : |
| IsLocalRing.maximalIdeal E.LocalRing = |
| (MvPolynomial.idealOfVars α K).map (algebraMap (MvPolynomial α K) E.LocalRing) := by |
| rw [maximalIdeal_eq_map_origin,origin_eq_variables] |
|
|
| def augmentation (E : AffineEtaleChart K α) : E.LocalRing →+* K := |
| constantCoeff.comp E.localExpansion.toRingHom |
|
|
| @[simp] lemma augmentation_polynomial (E : AffineEtaleChart K α) (p : MvPolynomial α K) : |
| E.augmentation (algebraMap (MvPolynomial α K) E.LocalRing p) = |
| MvPolynomial.constantCoeff p := by |
| change constantCoeff (E.localExpansion (algebraMap (MvPolynomial α K) E.LocalRing p)) = _ |
| rw [E.localExpansion.commutes] |
| rfl |
|
|
| lemma augmentation_ker (E : AffineEtaleChart K α) : |
| RingHom.ker E.augmentation = IsLocalRing.maximalIdeal E.LocalRing := by |
| ext a |
| change E.augmentation a = 0 ↔ _ |
| rw [IsLocalRing.mem_maximalIdeal] |
| constructor |
| · intro h hu |
| have hh := hu.map E.augmentation |
| rw [h] at hh |
| exact not_isUnit_zero hh |
| · intro h |
| by_contra hn |
| apply h |
| apply isUnit_of_map_unit E.localExpansion |
| rw [MvPowerSeries.isUnit_iff_constantCoeff,isUnit_iff_ne_zero] |
| exact hn |
|
|
| abbrev FormalTruncation (q : ℕ) := TaylorTarget.Ring K α q |
|
|
| noncomputable def formalTruncate (q : ℕ) : |
| MvPowerSeries α K →ₐ[MvPolynomial α K] FormalTruncation (K := K) (α := α) q := |
| MvPowerSeries.truncTotalAlgHom α K q |
|
|
| lemma reduction_formalTruncate (q : ℕ) (hq : 1 ≤ q) (f : MvPowerSeries α K) : |
| TaylorTarget.reduction K α q hq (formalTruncate q f) = constantCoeff f := by |
| change MvPolynomial.constantCoeff (MvPowerSeries.truncTotal q f) = constantCoeff f |
| exact MvPowerSeries.coeff_truncTotal _ (by simp only [map_zero]; omega) |
|
|
| noncomputable def truncate (E : AffineEtaleChart K α) (q : ℕ) : |
| E.LocalRing →ₐ[MvPolynomial α K] FormalTruncation (K := K) (α := α) q := |
| (formalTruncate q).comp E.localExpansion |
|
|
| @[simp] lemma truncate_polynomial (E : AffineEtaleChart K α) (q : ℕ) (p : MvPolynomial α K) : |
| E.truncate q (algebraMap (MvPolynomial α K) E.LocalRing p) = |
| Ideal.Quotient.mk (MvPolynomial.idealOfVars α K ^ q) p := |
| (E.truncate q).commutes p |
|
|
| lemma reduction_truncate (E : AffineEtaleChart K α) (q : ℕ) (hq : 1 ≤ q) (a : E.LocalRing) : |
| TaylorTarget.reduction K α q hq (E.truncate q a) = E.augmentation a := |
| reduction_formalTruncate q hq _ |
|
|
| lemma truncate_maximal_pow (E : AffineEtaleChart K α) (q : ℕ) : |
| (IsLocalRing.maximalIdeal E.LocalRing) ^ q ≤ RingHom.ker (E.truncate q) := by |
| rw [E.maximalIdeal_eq_map_variables,← Ideal.map_pow,Ideal.map_le_iff_le_comap] |
| intro p hp |
| change E.truncate q (algebraMap (MvPolynomial α K) E.LocalRing p) = 0 |
| rw [truncate_polynomial,Ideal.Quotient.eq_zero_iff_mem] |
| exact hp |
|
|
| noncomputable def truncateQuotient (E : AffineEtaleChart K α) (q : ℕ) : |
| E.LocalRing ⧸ (IsLocalRing.maximalIdeal E.LocalRing) ^ q →+* |
| FormalTruncation (K := K) (α := α) q := |
| Ideal.Quotient.lift _ (E.truncate q).toRingHom (E.truncate_maximal_pow q) |
|
|
| noncomputable def polynomialQuotient (E : AffineEtaleChart K α) (q : ℕ) : |
| FormalTruncation (K := K) (α := α) q →ₐ[MvPolynomial α K] |
| E.LocalRing ⧸ (IsLocalRing.maximalIdeal E.LocalRing) ^ q := |
| Ideal.Quotient.liftₐ _ (Algebra.ofId (MvPolynomial α K) (E.LocalRing ⧸ (IsLocalRing.maximalIdeal E.LocalRing) ^ q)) (by |
| intro p hp |
| change Ideal.Quotient.mk _ (algebraMap (MvPolynomial α K) E.LocalRing p) = 0 |
| rw [Ideal.Quotient.eq_zero_iff_mem,E.maximalIdeal_eq_map_variables,← Ideal.map_pow] |
| exact Ideal.mem_map_of_mem _ hp) |
|
|
| lemma truncate_polynomialQuotient (E : AffineEtaleChart K α) (q : ℕ) |
| (x : FormalTruncation (K := K) (α := α) q) : |
| E.truncateQuotient q (E.polynomialQuotient q x) = x := by |
| obtain ⟨p,rfl⟩ := Ideal.Quotient.mk_surjective x |
| change E.truncate q (algebraMap (MvPolynomial α K) E.LocalRing p) = _ |
| exact E.truncate_polynomial q p |
|
|
| lemma factor_polynomial_truncate (E : AffineEtaleChart K α) (q : ℕ) (hq : 1 ≤ q) |
| (a : E.LocalRing) : |
| Ideal.Quotient.factor (Ideal.pow_le_self (show q ≠ 0 by omega)) |
| (E.polynomialQuotient q (E.truncate q a)) = |
| Ideal.Quotient.mk (IsLocalRing.maximalIdeal E.LocalRing) a := by |
| obtain ⟨p,hp⟩ := Ideal.Quotient.mk_surjective (E.truncate q a) |
| have hc : MvPolynomial.constantCoeff p = E.augmentation a := by |
| have hh := congrArg (TaylorTarget.reduction K α q hq) hp |
| exact (TaylorTarget.reduction_mk K α q hq p).symm.trans |
| (hh.trans (E.reduction_truncate q hq a)) |
| rw [← hp] |
| change Ideal.Quotient.mk (IsLocalRing.maximalIdeal E.LocalRing) |
| (algebraMap (MvPolynomial α K) E.LocalRing p) = |
| Ideal.Quotient.mk (IsLocalRing.maximalIdeal E.LocalRing) a |
| rw [Ideal.Quotient.eq,← E.augmentation_ker] |
| change E.augmentation (algebraMap (MvPolynomial α K) E.LocalRing p - a) = 0 |
| rw [map_sub,augmentation_polynomial,hc,sub_self] |
|
|
| lemma polynomial_truncate (E : AffineEtaleChart K α) (q : ℕ) (a : E.LocalRing) : |
| E.polynomialQuotient q (E.truncate q a) = |
| Ideal.Quotient.mk ((IsLocalRing.maximalIdeal E.LocalRing)^q) a := by |
| by_cases hq : q = 0 |
| · subst q |
| have : Subsingleton (E.LocalRing ⧸ (IsLocalRing.maximalIdeal E.LocalRing)^0) := |
| Ideal.Quotient.subsingleton_iff.mpr (by simp) |
| exact Subsingleton.elim _ _ |
| · let r := Ideal.Quotient.factor |
| (Ideal.pow_le_self (I := IsLocalRing.maximalIdeal E.LocalRing) hq) |
| have hr : IsNilpotent (RingHom.ker r) := by |
| refine ⟨q,?_⟩ |
| have hk : RingHom.ker r = (IsLocalRing.maximalIdeal E.LocalRing).map |
| (Ideal.Quotient.mk ((IsLocalRing.maximalIdeal E.LocalRing)^q)) := by |
| apply Ideal.comap_injective_of_surjective |
| (Ideal.Quotient.mk ((IsLocalRing.maximalIdeal E.LocalRing)^q)) Ideal.Quotient.mk_surjective |
| rw [RingHom.comap_ker] |
| change RingHom.ker (Ideal.Quotient.mk (IsLocalRing.maximalIdeal E.LocalRing)) = _ |
| rw [Ideal.mk_ker,Ideal.comap_map_of_surjective _ Ideal.Quotient.mk_surjective] |
| change _ = IsLocalRing.maximalIdeal E.LocalRing ⊔ _ |
| rw [← RingHom.ker_eq_comap_bot,Ideal.mk_ker] |
| exact (sup_eq_left.mpr (Ideal.pow_le_self hq)).symm |
| rw [hk,← Ideal.map_pow,Ideal.map_quotient_self] |
| rfl |
| have hh := Algebra.FormallyUnramified.ext' r hr |
| ((E.polynomialQuotient q).comp (E.truncate q)) |
| (Ideal.Quotient.mkₐ (MvPolynomial α K) ((IsLocalRing.maximalIdeal E.LocalRing)^q)) |
| (fun a ↦ E.factor_polynomial_truncate q (by omega) a) |
| exact AlgHom.congr_fun hh a |
|
|
| noncomputable def truncatedEquiv (E : AffineEtaleChart K α) (q : ℕ) : |
| E.LocalRing ⧸ (IsLocalRing.maximalIdeal E.LocalRing)^q ≃+* |
| FormalTruncation (K := K) (α := α) q := |
| { E.truncateQuotient q with |
| invFun := E.polynomialQuotient q |
| left_inv := by |
| intro x |
| obtain ⟨a,rfl⟩ := Ideal.Quotient.mk_surjective x |
| exact E.polynomial_truncate q a |
| right_inv := E.truncate_polynomialQuotient q } |
|
|
| @[simp] lemma truncatedEquiv_mk (E : AffineEtaleChart K α) (q : ℕ) (a : E.LocalRing) : |
| E.truncatedEquiv q (Ideal.Quotient.mk _ a) = E.truncate q a := rfl |
|
|
| end BoundaryOnly.FormalObstruction.FormalCorrection.AffineEtaleChart |
|
|
| end |
|
|
| end OAI |
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|
