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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 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 | import Mathlib
import OAI.Algebra.FrobeniusObstruction.Obstruction
import OAI.Algebra.AlgebraicObstruction.ChartTruncation
import OAI.Algebra.AlgebraicObstruction.AdicFunctor
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]
lemma truncatedEquiv_compatible (E : AffineEtaleChart K α) {m n : ℕ} (h : m ≤ n)
(x : E.LocalRing ⧸ (IsLocalRing.maximalIdeal E.LocalRing)^n) :
Ideal.Quotient.factorPow (MvPolynomial.idealOfVars α K) h (E.truncatedEquiv n x) =
E.truncatedEquiv m (Ideal.Quotient.factorPow (IsLocalRing.maximalIdeal E.LocalRing) h x) := by
obtain ⟨a,rfl⟩ := Ideal.Quotient.mk_surjective x
change Ideal.Quotient.mk (MvPolynomial.idealOfVars α K ^ m)
(truncTotal n (E.localExpansion a)) =
Ideal.Quotient.mk (MvPolynomial.idealOfVars α K ^ m)
(truncTotal m (E.localExpansion a))
rw [Ideal.Quotient.eq]
exact truncTotal_sub_truncTotal_mem_pow_idealOfVars h le_rfl _
def completionFamily (E : AffineEtaleChart K α) (n : ℕ) :
AdicCompletion (IsLocalRing.maximalIdeal E.LocalRing) E.LocalRing →+*
FormalTruncation (K := K) (α := α) n :=
(E.truncatedEquiv n).toRingHom.comp
(AdicCompletion.evalₐ (IsLocalRing.maximalIdeal E.LocalRing) n).toRingHom
lemma completionFamily_compatible (E : AffineEtaleChart K α) {m n : ℕ} (h : m ≤ n) :
(Ideal.Quotient.factorPow (MvPolynomial.idealOfVars α K) h).comp (E.completionFamily n) =
E.completionFamily m := by
apply RingHom.ext
intro x
change Ideal.Quotient.factorPow _ h (E.truncatedEquiv n (AdicCompletion.evalₐ _ n x)) = _
rw [E.truncatedEquiv_compatible h,AdicCofinal.factor_eval]
rfl
noncomputable def completionToPolynomial (E : AffineEtaleChart K α) :
AdicCompletion (IsLocalRing.maximalIdeal E.LocalRing) E.LocalRing →+*
AdicCompletion (MvPolynomial.idealOfVars α K) (MvPolynomial α K) :=
AdicCompletion.liftRingHom _ E.completionFamily E.completionFamily_compatible
@[simp] lemma eval_completionToPolynomial (E : AffineEtaleChart K α) (n : ℕ)
(x : AdicCompletion (IsLocalRing.maximalIdeal E.LocalRing) E.LocalRing) :
AdicCompletion.evalₐ _ n (E.completionToPolynomial x) =
E.truncatedEquiv n (AdicCompletion.evalₐ _ n x) :=
AdicCompletion.evalₐ_liftRingHom _ E.completionFamily E.completionFamily_compatible n x
omit [Finite α] in
lemma variables_le_comap_maximal (E : AffineEtaleChart K α) :
MvPolynomial.idealOfVars α K ≤ (IsLocalRing.maximalIdeal E.LocalRing).comap
(algebraMap (MvPolynomial α K) E.LocalRing) := by
rw [E.maximalIdeal_eq_map_variables]
exact Ideal.le_comap_map
noncomputable def polynomialToCompletion (E : AffineEtaleChart K α) :
AdicCompletion (MvPolynomial.idealOfVars α K) (MvPolynomial α K) →+*
AdicCompletion (IsLocalRing.maximalIdeal E.LocalRing) E.LocalRing :=
AdicFunctor.map _ _ (algebraMap (MvPolynomial α K) E.LocalRing) E.variables_le_comap_maximal
omit [Finite α] in
lemma quotient_polynomial (E : AffineEtaleChart K α) (n : ℕ)
(x : FormalTruncation (K := K) (α := α) n) :
AdicFunctor.quotient _ _ (algebraMap (MvPolynomial α K) E.LocalRing)
E.variables_le_comap_maximal n x = E.polynomialQuotient n x := by
obtain ⟨p,rfl⟩ := Ideal.Quotient.mk_surjective x
rfl
omit [Finite α] in
@[simp] lemma eval_polynomialToCompletion (E : AffineEtaleChart K α) (n : ℕ)
(x : AdicCompletion (MvPolynomial.idealOfVars α K) (MvPolynomial α K)) :
AdicCompletion.evalₐ _ n (E.polynomialToCompletion x) =
E.polynomialQuotient n (AdicCompletion.evalₐ _ n x) := by
rw [polynomialToCompletion,AdicFunctor.eval_map,quotient_polynomial]
lemma completionToPolynomial_polynomialToCompletion (E : AffineEtaleChart K α)
(x : AdicCompletion (MvPolynomial.idealOfVars α K) (MvPolynomial α K)) :
E.completionToPolynomial (E.polynomialToCompletion x) = x := by
apply AdicCompletion.ext_evalₐ
intro n
rw [eval_completionToPolynomial,eval_polynomialToCompletion]
exact (E.truncatedEquiv n).apply_symm_apply _
lemma polynomialToCompletion_completionToPolynomial (E : AffineEtaleChart K α)
(x : AdicCompletion (IsLocalRing.maximalIdeal E.LocalRing) E.LocalRing) :
E.polynomialToCompletion (E.completionToPolynomial x) = x := by
apply AdicCompletion.ext_evalₐ
intro n
rw [eval_polynomialToCompletion,eval_completionToPolynomial]
exact (E.truncatedEquiv n).symm_apply_apply _
noncomputable def polynomialCompletionEquiv (E : AffineEtaleChart K α) :
AdicCompletion (IsLocalRing.maximalIdeal E.LocalRing) E.LocalRing ≃+*
AdicCompletion (MvPolynomial.idealOfVars α K) (MvPolynomial α K) :=
{ E.completionToPolynomial with
invFun := E.polynomialToCompletion
left_inv := E.polynomialToCompletion_completionToPolynomial
right_inv := E.completionToPolynomial_polynomialToCompletion }
noncomputable def completionEquiv (E : AffineEtaleChart K α) :
AdicCompletion (IsLocalRing.maximalIdeal E.LocalRing) E.LocalRing ≃+*
MvPowerSeries α K :=
E.polynomialCompletionEquiv.trans (MvPowerSeries.toAdicCompletionAlgEquiv α K).symm.toRingEquiv
@[simp] theorem completionEquiv_of (E : AffineEtaleChart K α) (a : E.LocalRing) :
E.completionEquiv (AdicCompletion.of _ _ a) = E.localExpansion a := by
apply (MvPowerSeries.toAdicCompletionAlgEquiv α K).injective
change (MvPowerSeries.toAdicCompletionAlgEquiv α K)
((MvPowerSeries.toAdicCompletionAlgEquiv α K).symm
(E.completionToPolynomial (AdicCompletion.of _ _ a))) = _
rw [AlgEquiv.apply_symm_apply]
apply AdicCompletion.ext_evalₐ
intro n
rw [eval_completionToPolynomial,AdicCompletion.evalₐ_of,truncatedEquiv_mk]
rfl
noncomputable def completionAlgEquiv (E : AffineEtaleChart K α) :
AdicCompletion (IsLocalRing.maximalIdeal E.LocalRing) E.LocalRing ≃ₐ[E.LocalRing]
MvPowerSeries α K :=
{ E.completionEquiv with commutes' := E.completionEquiv_of }
instance localNoetherian (E : AffineEtaleChart K α) : IsNoetherianRing E.LocalRing := by
have : IsNoetherianRing E.S :=
Algebra.FiniteType.isNoetherianRing (MvPolynomial α K) E.S
infer_instance
end BoundaryOnly.FormalObstruction.FormalCorrection.AffineEtaleChart
end
end OAI
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