File size: 3,220 Bytes
7f5fab8 | 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 | 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)))
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
end OAI
|