Datasets:

roseking
/

openAI-math-dataset

import OAI.AlgebraicGeometry.CartierSections.ChartRetraction
import OAI.AlgebraicGeometry.CartierSections.FractionValuation

namespace OAI

/-!
# Monomial retractions on rational function fields

The closed étale chart retraction extends to the chart's fraction field.
-/

noncomputable section
open scoped BigOperators NNReal ENNReal
namespace CartierSections
section ClosedRetractionField
universe u
local instance polynomialOrigin_isMaximal_field {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]

/-- The retraction is a valuation of the chart's actual rational function field. -/
noncomputable def closedChartFieldRetraction (w : σ → NNReal) : AddValuation K (WithTop ℝ) :=
  fractionValuation (closedChartRetraction (σ := σ) (k := k) (A := A) (R := R) w)
    (fun _ hf => (closedChartRetraction_finite (σ := σ) (k := k) (A := A) (R := R) w hf).ne)

@[simp] lemma closedChartFieldRetraction_regular (w : σ → NNReal) (f : A) :
    closedChartFieldRetraction (σ := σ) (k := k) (R := R) (A := A) (K := K) w (algebraMap A K f) =
      nonnegToExtendedReal (closedChartRetraction (R := R) (k := k) w f) :=
  fractionValuation_algebraMap _ _ _

end ClosedRetractionField
end CartierSections

end

end OAI

	import OAI.AlgebraicGeometry.CartierSections.ChartRetraction
	import OAI.AlgebraicGeometry.CartierSections.FractionValuation

	namespace OAI

	/-!
	# Monomial retractions on rational function fields

	The closed étale chart retraction extends to the chart's fraction field.
	-/

	noncomputable section
	open scoped BigOperators NNReal ENNReal
	namespace CartierSections
	section ClosedRetractionField
	universe u
	local instance polynomialOrigin_isMaximal_field {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]

	/-- The retraction is a valuation of the chart's actual rational function field. -/
	noncomputable def closedChartFieldRetraction (w : σ → NNReal) : AddValuation K (WithTop ℝ) :=
	fractionValuation (closedChartRetraction (σ := σ) (k := k) (A := A) (R := R) w)
	(fun _ hf => (closedChartRetraction_finite (σ := σ) (k := k) (A := A) (R := R) w hf).ne)

	@[simp] lemma closedChartFieldRetraction_regular (w : σ → NNReal) (f : A) :
	closedChartFieldRetraction (σ := σ) (k := k) (R := R) (A := A) (K := K) w (algebraMap A K f) =
	nonnegToExtendedReal (closedChartRetraction (R := R) (k := k) w f) :=
	fractionValuation_algebraMap _ _ _

	end ClosedRetractionField
	end CartierSections

	end

	end OAI