of_isLocalization — Mathlib

English

For a prime p and localization S at p with appropriate hypotheses, R localized at p has Krull dimension ≤ 0.

Русский

Для произвольного p и локализации S в p размер Крull ≤ 0.

LaTeX

$$$\\text{IsLocalization.AtPrime } S p \\Rightarrow Ring.KrullDimLE 0 S$$$

Lean4

theorem of_isLocalization (p : Ideal R) (hp : p ∈ minimalPrimes R) (S : Type*) [CommSemiring S] [Algebra R S]
    [IsLocalization.AtPrime S p (hp := hp.1.1)] : Ring.KrullDimLE 0 S :=
  have := IsLocalization.subsingleton_primeSpectrum_of_mem_minimalPrimes p hp S
  ⟨Order.krullDim_nonpos_of_subsingleton⟩

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