Theorems · Definition · commutative algebra

PrimeSpectrum.piLocalizationToMaximalEquiv

{R : Type u_1} →
  [inst : CommSemiring R] →
    (∀ (I : Ideal R), I.IsPrime → I.IsMaximal) → PrimeSpectrum.PiLocalization R ≃+* MaximalSpectrum.PiLocalization R

If R has Krull dimension ≤ 0, then piLocalizationToIsMaximal R is an isomorphism.

Defined in: Mathlib.RingTheory.Spectrum.Maximal.Localization
Cited by: 1 results in Mathlib
Foundations: Depth 74 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CommSemiring

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Cites23

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

DFunLike.coeproof · cited by 62,936
CommSemiringstatement and proof · cited by 10,911
Idealstatement and proof · cited by 4,748
AlgHomproof · cited by 3,236
RingEquivstatement · cited by 1,147
Ideal.IsPrimestatement and proof · cited by 827
PrimeSpectrumstatement and proof · cited by 625
AlgHom.toRingHomproof · cited by 490
Ideal.primeComplstatement · cited by 462
Ideal.IsMaximalstatement and proof · cited by 452
PrimeSpectrum.asIdealstatement · cited by 333
Localization.AtPrimestatement and proof · cited by 299

Cited by1

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PrimeSpectrum.piLocalizationToMaximal_bijectiveproof · cited by 0