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cfa2fd2 | 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 | 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
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