| import Mathlib |
| import OAI.Algebra.FrobeniusObstruction.Obstruction |
| import OAI.Algebra.AlgebraicObstruction.EtaleCharts |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped BigOperators |
|
|
| namespace BoundaryOnly.FormalObstruction.FormalCorrection.AffineEtaleChart |
| open MvPowerSeries |
| variable {K α : Type} [Field K] [Finite α] |
|
|
| def origin : Ideal (MvPolynomial α K) := RingHom.ker MvPolynomial.constantCoeff |
| instance : (origin (K := K) (α := α)).IsPrime := RingHom.ker_isPrime _ |
| |
| instance point_liesOver (E : AffineEtaleChart K α) : |
| E.point.LiesOver (origin (K := K) (α := α)) := by |
| constructor |
| ext p |
| change MvPolynomial.constantCoeff p = 0 ↔ |
| constantCoeff (E.expansion (algebraMap (MvPolynomial α K) E.S p)) = 0 |
| rw [E.expansion.commutes] |
| rfl |
| |
| noncomputable instance originAlgebra (E : AffineEtaleChart K α) : |
| Algebra (Localization.AtPrime (origin (K := K) (α := α))) E.LocalRing := |
| Localization.AtPrime.algebraOfLiesOver _ _ |
| |
| instance originLiesOverAlgebra (E : AffineEtaleChart K α) : |
| IsScalarTower (MvPolynomial α K) |
| (Localization.AtPrime (origin (K := K) (α := α))) E.LocalRing := |
| inferInstance |
| |
| instance origin_isLocalHom (E : AffineEtaleChart K α) : |
| IsLocalHom (algebraMap (Localization.AtPrime (origin (K := K) (α := α))) E.LocalRing) := by |
| change IsLocalHom (Localization.localRingHom _ _ (algebraMap (MvPolynomial α K) E.S) _) |
| infer_instance |
| |
| omit [Finite α] in |
| lemma maximalIdeal_eq_map_origin (E : AffineEtaleChart K α) : |
| IsLocalRing.maximalIdeal E.LocalRing = |
| (origin (K := K) (α := α)).map (algebraMap (MvPolynomial α K) E.LocalRing) := by |
| let R := Localization.AtPrime (origin (K := K) (α := α)) |
| have : Algebra.EssFiniteType R E.LocalRing := |
| Algebra.EssFiniteType.of_comp (MvPolynomial α K) R E.LocalRing |
| have : Algebra.FormallyUnramified R E.LocalRing := |
| Algebra.FormallyUnramified.of_restrictScalars (MvPolynomial α K) R E.LocalRing |
| rw [← Algebra.FormallyUnramified.map_maximalIdeal (R := R)] |
| rw [← Localization.AtPrime.map_eq_maximalIdeal (I := origin (K := K) (α := α))] |
| rw [Ideal.map_map,← IsScalarTower.algebraMap_eq (MvPolynomial α K) R E.LocalRing] |
| |
| end BoundaryOnly.FormalObstruction.FormalCorrection.AffineEtaleChart |
| |
| end |
| |
| end OAI |
| |
