| /- |
| Copyright (c) 2026 OpenAI. All rights reserved. |
| Released under Apache 2.0 license as described in the file LICENSE. |
| Authors: OpenAI |
| -/ |
| import OAI.AlgebraicGeometry.CartierSections.InfinitesimalCharts |
| import Mathlib.RingTheory.AdicCompletion.AsTensorProduct |
| import Mathlib.RingTheory.AdicCompletion.LocalRing |
| import Mathlib.RingTheory.Flat.FaithfullyFlat.Algebra |
| import Mathlib.RingTheory.MvPowerSeries.Equiv |
|
|
| namespace OAI |
|
|
| /-! |
| |
|
|
| Compatible infinitesimal isomorphisms identify an étale chart completion with a |
| power-series ring. Expansion through a Noetherian local completion is faithfully flat. |
| -/ |
|
|
| namespace CartierSections |
|
|
| attribute [local instance 10001] scalarTower_self |
| section CompletionIsomorphism |
| variable {R A : Type*} [CommRing R] [CommRing A] [Algebra R A] |
|
|
| /-- Evaluation of an adic completion commutes with reduction to a lower ideal power. -/ |
| theorem completion_eval_factor (I : Ideal R) {m n : ℕ} (h : m ≤ n) |
| (x : AdicCompletion I R) : |
| Ideal.Quotient.factorₐ R (Ideal.pow_le_pow_right h) (AdicCompletion.evalₐ I n x) = |
| AdicCompletion.evalₐ I m x := by |
| have hx := congrArg (Ideal.Quotient.factor (show (I^m • ⊤ : Ideal R) ≤ I^m |
| from le_of_eq (Ideal.mul_top _))) (x.property h) |
| simpa [AdicCompletion.evalₐ, AdicCompletion.eval, |
| AdicCompletion.transitionMap, Submodule.factorPow] using hx |
|
|
| /-- Evaluation on infinitesimal neighbourhoods is compatible with the maps induced |
| by an algebra map. -/ |
| theorem quotientPowerMap_eval_compatible (I : Ideal R) {m n : ℕ} (h : m ≤ n) : |
| (Ideal.Quotient.factorₐ R |
| (Ideal.pow_le_pow_right (I := I.map (algebraMap R A)) h)).comp |
| ((quotientPowerMap I n).comp (AdicCompletion.evalₐ I n)) = |
| (quotientPowerMap I m).comp (AdicCompletion.evalₐ I m) := by |
| rw [← AlgHom.comp_assoc, quotientPowerMap_factor I h, AlgHom.comp_assoc] |
| congr 1 |
| ext x |
| exact completion_eval_factor I h x |
|
|
| /-- The completion map induced by an algebra map. -/ |
| noncomputable def chartCompletionMap (I : Ideal R) : |
| AdicCompletion I R →ₐ[R] AdicCompletion (I.map (algebraMap R A)) A := |
| AdicCompletion.liftAlgHom _ |
| (fun n => (quotientPowerMap I n).comp (AdicCompletion.evalₐ I n)) |
| (quotientPowerMap_eval_compatible I) |
|
|
| @[simp] theorem chartCompletionMap_eval (I : Ideal R) (n : ℕ) (x : AdicCompletion I R) : |
| AdicCompletion.evalₐ (I.map (algebraMap R A)) n (chartCompletionMap I x) = |
| quotientPowerMap I n (AdicCompletion.evalₐ I n x) := |
| AdicCompletion.evalₐ_liftAlgHom _ _ (quotientPowerMap_eval_compatible I) n x |
|
|
| /-- Étale charts with trivial residue extension have the same completion. |
| This is derived from formal lifting and uniqueness at every order. -/ |
| theorem chartCompletionMap_bijective [Algebra.FormallyEtale R A] |
| (I : Ideal R) (hI : Function.Bijective (quotientBaseMap (A := A) I)) : |
| Function.Bijective (chartCompletionMap (A := A) I) := by |
| classical |
| let J := I.map (algebraMap R A) |
| let e (n : ℕ) := AlgEquiv.ofBijective (quotientPowerMap (A := A) I n) |
| (etale_quotient_power_bijective I hI n) |
| let g (n : ℕ) : AdicCompletion J A →ₐ[R] R ⧸ I^n := |
| (e n).symm.toAlgHom.comp ((AdicCompletion.evalₐ J n).restrictScalars R) |
| have hg {m n : ℕ} (h : m ≤ n) : |
| (Ideal.Quotient.factorₐ R (Ideal.pow_le_pow_right h)).comp (g n) = g m := by |
| ext x |
| apply (e m).injective |
| change quotientPowerMap I m |
| (Ideal.Quotient.factorₐ R (Ideal.pow_le_pow_right h) ((e n).symm (AdicCompletion.evalₐ J n x))) = _ |
| have hfac := AlgHom.congr_fun (quotientPowerMap_factor (A := A) I h) |
| ((e n).symm (AdicCompletion.evalₐ J n x)) |
| simp only [AlgHom.comp_apply] at hfac |
| rw [← hfac] |
| change Ideal.Quotient.factorₐ R (Ideal.pow_le_pow_right h) |
| (e n ((e n).symm (AdicCompletion.evalₐ J n x))) = e m ((e m).symm (AdicCompletion.evalₐ J m x)) |
| rw [(e n).apply_symm_apply, (e m).apply_symm_apply] |
| exact completion_eval_factor J h x |
| let inv : AdicCompletion J A →ₐ[R] AdicCompletion I R := |
| AdicCompletion.liftAlgHom I g (fun h => hg h) |
| have hleft : Function.LeftInverse inv (chartCompletionMap (A := A) I) := by |
| intro x |
| apply AdicCompletion.ext_evalₐ |
| intro n |
| rw [AdicCompletion.evalₐ_liftAlgHom] |
| change (e n).symm (AdicCompletion.evalₐ J n (chartCompletionMap I x)) = _ |
| rw [chartCompletionMap_eval] |
| exact (e n).symm_apply_apply _ |
| have hright : Function.RightInverse inv (chartCompletionMap (A := A) I) := by |
| intro x |
| apply AdicCompletion.ext_evalₐ |
| intro n |
| rw [chartCompletionMap_eval, AdicCompletion.evalₐ_liftAlgHom] |
| exact (e n).apply_symm_apply _ |
| exact ⟨hleft.injective, hright.surjective⟩ |
|
|
| noncomputable def chartCompletionEquiv [Algebra.FormallyEtale R A] |
| (I : Ideal R) (hI : Function.Bijective (quotientBaseMap (A := A) I)) : |
| AdicCompletion I R ≃ₐ[R] AdicCompletion (I.map (algebraMap R A)) A := |
| AlgEquiv.ofBijective (chartCompletionMap I) (chartCompletionMap_bijective I hI) |
| end CompletionIsomorphism |
| /-- Formal coordinates on the completion of an étale polynomial chart. -/ |
| noncomputable def polynomialChartSeriesEquiv |
| {σ k A : Type*} [Finite σ] [CommRing k] [CommRing A] |
| [Algebra (MvPolynomial σ k) A] [Algebra.FormallyEtale (MvPolynomial σ k) A] |
| (hI : Function.Bijective |
| (quotientBaseMap (A := A) (MvPolynomial.idealOfVars σ k))) : |
| MvPowerSeries σ k ≃ₐ[MvPolynomial σ k] |
| AdicCompletion ((MvPolynomial.idealOfVars σ k).map (algebraMap (MvPolynomial σ k) A)) A := |
| (MvPowerSeries.toAdicCompletionAlgEquiv σ k).trans |
| (chartCompletionEquiv (MvPolynomial.idealOfVars σ k) hI) |
|
|
| @[simp] theorem polynomialChartSeriesEquiv_coe |
| {σ k A : Type*} [Finite σ] [CommRing k] [CommRing A] |
| [Algebra (MvPolynomial σ k) A] [Algebra.FormallyEtale (MvPolynomial σ k) A] |
| (hI : Function.Bijective |
| (quotientBaseMap (A := A) (MvPolynomial.idealOfVars σ k))) (p : MvPolynomial σ k) : |
| polynomialChartSeriesEquiv hI (p : MvPowerSeries σ k) = |
| algebraMap A _ (algebraMap (MvPolynomial σ k) A p) := by |
| exact (polynomialChartSeriesEquiv hI).commutes p |
| /-- The expansion map attached to an isomorphism with the maximal-ideal completion. -/ |
| noncomputable def completionExpansion |
| {A S : Type*} [CommRing A] [IsLocalRing A] [CommRing S] |
| (e : S ≃+* AdicCompletion (IsLocalRing.maximalIdeal A) A) : A →+* S := |
| e.symm.toRingHom.comp (algebraMap A _) |
|
|
| /-- Formal expansion through a Noetherian completion is faithfully flat. -/ |
| theorem completionExpansion_faithfullyFlat |
| {A S : Type*} [CommRing A] [IsLocalRing A] [IsNoetherianRing A] [CommRing S] |
| (e : S ≃+* AdicCompletion (IsLocalRing.maximalIdeal A) A) : |
| letI : Algebra A S := (completionExpansion e).toAlgebra |
| Module.FaithfullyFlat A S := by |
| let : Algebra A S := (completionExpansion e).toAlgebra |
| have : Module.FaithfullyFlat A (AdicCompletion (IsLocalRing.maximalIdeal A) A) := |
| Module.FaithfullyFlat.of_flat_of_isLocalHom |
| let eA : S ≃ₐ[A] AdicCompletion (IsLocalRing.maximalIdeal A) A := |
| { e with commutes' := fun a => e.apply_symm_apply (algebraMap A _ a) } |
| exact Module.FaithfullyFlat.of_linearEquiv A _ eA.toLinearEquiv |
|
|
| /-- A formal expansion through a Noetherian local completion reflects units. -/ |
| theorem completionExpansion_isLocalHom |
| {A S : Type*} [CommRing A] [IsLocalRing A] [IsNoetherianRing A] [CommRing S] |
| (e : S ≃+* AdicCompletion (IsLocalRing.maximalIdeal A) A) : |
| IsLocalHom (completionExpansion e) := by |
| constructor |
| intro a ha |
| apply (isUnit_map_iff (algebraMap A (AdicCompletion (IsLocalRing.maximalIdeal A) A)) a).mp |
| have h := ha.map e.toMonoidHom |
| simpa [completionExpansion] using h |
| section ChartExpansion |
| variable {k A σ : Type*} [CommRing k] [CommRing A] [IsLocalRing A] |
| [Algebra k A] [Algebra (MvPolynomial σ k) A] [IsScalarTower k (MvPolynomial σ k) A] |
|
|
| /-- Expansion in completion coordinates as an algebra map over the coefficient ring. -/ |
| noncomputable def chartSeriesExpansion |
| (e : MvPowerSeries σ k ≃ₐ[MvPolynomial σ k] |
| AdicCompletion (IsLocalRing.maximalIdeal A) A) : A →ₐ[k] MvPowerSeries σ k := |
| (e.symm.restrictScalars k).toAlgHom.comp |
| (IsScalarTower.toAlgHom k A (AdicCompletion (IsLocalRing.maximalIdeal A) A)) |
|
|
| @[simp] theorem chartSeriesExpansion_toRingHom |
| (e : MvPowerSeries σ k ≃ₐ[MvPolynomial σ k] |
| AdicCompletion (IsLocalRing.maximalIdeal A) A) : |
| (chartSeriesExpansion e).toRingHom = completionExpansion e.toRingEquiv := rfl |
|
|
| @[simp] theorem chartSeriesExpansion_coordinate |
| (e : MvPowerSeries σ k ≃ₐ[MvPolynomial σ k] |
| AdicCompletion (IsLocalRing.maximalIdeal A) A) (i : σ) : |
| chartSeriesExpansion e (algebraMap (MvPolynomial σ k) A (MvPolynomial.X i)) = |
| MvPowerSeries.X i := by |
| apply e.injective |
| change e (e.symm (algebraMap A _ (algebraMap (MvPolynomial σ k) A (MvPolynomial.X i)))) = _ |
| rw [e.apply_symm_apply] |
| have he : e ((MvPolynomial.X i : MvPolynomial σ k) : MvPowerSeries σ k) = |
| algebraMap (MvPolynomial σ k) _ (MvPolynomial.X i) := by |
| simpa only [MvPowerSeries.algebraMap_apply', Algebra.algebraMap_self, |
| MvPowerSeries.map_id, RingHom.id_apply] using e.commutes (MvPolynomial.X i) |
| simpa only [MvPolynomial.coe_X, |
| IsScalarTower.algebraMap_apply (MvPolynomial σ k) A (AdicCompletion (IsLocalRing.maximalIdeal A) A)] using he.symm |
| end ChartExpansion |
| end CartierSections |
|
|
| end OAI |
|
|
