| import OAI.AlgebraicGeometry.CartierSections.ExpansionValuation |
| import OAI.AlgebraicGeometry.CartierSections.ClosedCharts |
|
|
| namespace OAI |
|
|
| /-! |
| # Monomial retractions on closed étale charts |
|
|
| The completed local coordinate isomorphism defines a faithful formal |
| expansion. Its monomial valuation allows boundary faces with zero weights. |
| -/ |
|
|
| noncomputable section |
| open scoped BigOperators NNReal ENNReal |
| namespace CartierSections |
| /-- Equal ideals have canonically isomorphic completions. -/ |
| noncomputable def completionIdealCongr {P A : Type*} [CommRing P] [CommRing A] |
| [Algebra P A] (I J : Ideal A) (h : I = J) : |
| AdicCompletion I A ≃ₐ[P] AdicCompletion J A := by |
| subst J |
| exact AlgEquiv.refl |
|
|
| section ClosedRetraction |
| attribute [local instance] polynomialOrigin_isMaximal |
| universe u |
| variable {σ k R A : Type u} [Fintype σ] [Field k] [IsAlgClosed k] |
| [CommRing R] [CommRing A] [IsLocalRing R] [IsLocalRing A] [IsNoetherianRing A] |
| [Algebra (MvPolynomial σ k) R] [Algebra (MvPolynomial σ k) A] |
| [Algebra k A] [IsScalarTower k (MvPolynomial σ k) A] |
| [Algebra R A] [IsScalarTower (MvPolynomial σ k) R A] |
| [IsLocalization.AtPrime R (MvPolynomial.idealOfVars σ k)] |
| [IsLocalHom (algebraMap R A)] [Algebra.FormallyUnramified R A] |
| [Algebra.EssFiniteType R A] [Algebra.FormallyEtale (MvPolynomial σ k) A] |
|
|
| include R |
|
|
| /-- The completed coordinate isomorphism at a closed étale point. -/ |
| noncomputable def closedChartEquiv : MvPowerSeries σ k ≃ₐ[MvPolynomial σ k] |
| AdicCompletion (IsLocalRing.maximalIdeal A) A := by |
| exact (polynomialChartSeriesEquiv |
| (closed_chart_base_bijective (R := R) (A := A) (σ := σ) (k := k))).trans |
| (completionIdealCongr _ _ |
| (closed_chart_maximalIdeal (R := R) (A := A) (σ := σ) (k := k))) |
|
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| noncomputable def closedChartExpansion : A →ₐ[k] MvPowerSeries σ k := |
| chartSeriesExpansion (closedChartEquiv (σ := σ) (k := k) (A := A) (R := R)) |
|
|
| omit [IsNoetherianRing A] in |
| @[simp] lemma closedChartExpansion_coordinate (i : σ) : |
| closedChartExpansion (σ := σ) (k := k) (A := A) (R := R) |
| (algebraMap (MvPolynomial σ k) A (MvPolynomial.X i)) = MvPowerSeries.X i := |
| chartSeriesExpansion_coordinate _ i |
|
|
| /-- The monomial valuation pulled back by the formal coordinate expansion. -/ |
| noncomputable def closedChartRetraction (w : σ → ℝ≥0) : AddValuation A ENNReal := |
| expansionValuation (closedChartExpansion (σ := σ) (k := k) (A := A) (R := R)) w |
|
|
| lemma closedChartExpansion_injective : |
| Function.Injective (closedChartExpansion (σ := σ) (k := k) (A := A) (R := R)) := by |
| let e := closedChartEquiv (σ := σ) (k := k) (A := A) (R := R) |
| let : Algebra A (MvPowerSeries σ k) := (completionExpansion e.toRingEquiv).toAlgebra |
| let : Module.FaithfullyFlat A (MvPowerSeries σ k) := |
| completionExpansion_faithfullyFlat e.toRingEquiv |
| exact FaithfulSMul.algebraMap_injective A (MvPowerSeries σ k) |
|
|
| lemma closedChartRetraction_finite (w : σ → ℝ≥0) {f : A} (hf : f ≠ 0) : |
| closedChartRetraction (σ := σ) (k := k) (A := A) (R := R) w f < ⊤ := |
| expansionValuation_finite _ (closedChartExpansion_injective (σ := σ) (k := k) (A := A) (R := R)) w hf |
|
|
| end ClosedRetraction |
| end CartierSections |
|
|
| end |
|
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| end OAI |
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|
