| 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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