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Copyright (c) 2026 OpenAI. All rights reserved.
Released under Apache 2.0 license.
Authors: OpenAI
-/
import OAI.AlgebraicGeometry.CartierSections.FieldTopology
import OAI.AlgebraicGeometry.CartierSections.FieldRetraction
namespace OAI
/-!
# Compactness of monomial retractions on a closed étale chart
Local algebra for uniform Cartier sections.
-/
noncomputable section
open scoped BigOperators NNReal ENNReal
namespace CartierSections
section ClosedChartFieldTopology
universe u
local instance polynomialOrigin_isMaximal_topology {σ k : Type u} [Fintype σ] [Field k] :
(MvPolynomial.idealOfVars σ k).IsMaximal :=
polynomialOrigin_isMaximal
variable {σ k R A K : Type u} [Fintype σ] [Field k] [IsAlgClosed k]
[CommRing R] [CommRing A] [IsLocalRing R] [IsLocalRing A] [IsNoetherianRing A]
[IsDomain A] [Field K] [Algebra A K] [IsFractionRing A K]
[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]
/-- Monomial retractions of a closed étale chart vary continuously with the weights. -/
lemma continuous_closedChartFieldRetraction_evaluations :
Continuous (fun w : σ → NNReal => fun x : K =>
closedChartFieldRetraction (σ := σ) (k := k) (R := R) (A := A) w x) :=
continuous_expandedFractionValuation_evaluations
(closedChartExpansion (σ := σ) (k := k) (R := R) (A := A))
(closedChartExpansion_injective (σ := σ) (k := k) (R := R) (A := A))
/-- The normalized monomial retractions on the fraction field of a closed étale
chart form a compact set in the topology of pointwise evaluation. -/
lemma isCompact_closedChartFieldRetraction_cone
(a : σ → NNReal) (ha : ∀ i, 0 < a i) :
IsCompact ((fun w : σ → NNReal => fun x : K =>
closedChartFieldRetraction (σ := σ) (k := k) (R := R) (A := A) w x) ''
normalizedWeights a) :=
(isCompact_normalizedWeights a ha).image
(continuous_closedChartFieldRetraction_evaluations
(σ := σ) (k := k) (R := R) (A := A) (K := K))
end ClosedChartFieldTopology
end CartierSections
end
end OAI
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