instEtaleOfIsStandardSmoothOfRelativeDimensionOfNatNat

English

The prime spectrum is in bijection with the set of maximal ideals via a natural equivalence mapping maximal ideals to their corresponding prime ideals and vice versa.

Русский

Спектр простых идеалов устанавливает биекцию с множеством максимальных идеалов через естественное отображение в обе стороны.

LaTeX

$$equivSubtype : MaximalSpectrum R ≃ { I : Ideal R // I.IsMaximal }$$

Lean4

/-- If `S` is `R`-standard smooth of relative dimension zero, it is étale. -/
instance (priority := 900) [IsStandardSmoothOfRelativeDimension 0 R S] : Etale R S
    where
  finitePresentation := (IsStandardSmoothOfRelativeDimension.isStandardSmooth 0).finitePresentation
  formallyEtale :=
    have : IsStandardSmooth R S := IsStandardSmoothOfRelativeDimension.isStandardSmooth 0
    have : FormallyUnramified R S := ⟨inferInstance⟩
    Algebra.FormallyEtale.of_unramified_and_smooth

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