Documentation

Mathlib.RingTheory.MaximalSpectrum

Maximal spectrum of a commutative ring #

The maximal spectrum of a commutative ring is the type of all maximal ideals. It is naturally a subset of the prime spectrum endowed with the subspace topology.

Main definitions #

Implementation notes #

The Zariski topology on the maximal spectrum is defined as the subspace topology induced by the natural inclusion into the prime spectrum to avoid API duplication for zero loci.

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theorem MaximalSpectrum.ext {R : Type u} :
∀ {inst : CommRing R} {x y : MaximalSpectrum R}, x.asIdeal = y.asIdeal → x = y
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theorem MaximalSpectrum.ext_iff {R : Type u} :
∀ {inst : CommRing R} {x y : MaximalSpectrum R}, x = y ↔ x.asIdeal = y.asIdeal
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structure MaximalSpectrum (R : Type u) [CommRing R] :
Type u

The maximal spectrum of a commutative ring R is the type of all maximal ideals of R.

  • asIdeal : Ideal R
  • IsMaximal : self.asIdeal.IsMaximal
Instances For
    source
    theorem MaximalSpectrum.IsMaximal {R : Type u} [CommRing R] (self : MaximalSpectrum R) :
    self.asIdeal.IsMaximal
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    instance MaximalSpectrum.instNonemptyOfNontrivial {R : Type u} [CommRing R] [Nontrivial R] :
    Nonempty (MaximalSpectrum R)
    Equations
    • ⋯ = ⋯
    source
    def MaximalSpectrum.toPrimeSpectrum {R : Type u} [CommRing R] (x : MaximalSpectrum R) :
    PrimeSpectrum R

    The natural inclusion from the maximal spectrum to the prime spectrum.

    Equations
    • x.toPrimeSpectrum = { asIdeal := x.asIdeal, isPrime := ⋯ }
    Instances For
      source
      theorem MaximalSpectrum.toPrimeSpectrum_injective {R : Type u} [CommRing R] :
      Function.Injective MaximalSpectrum.toPrimeSpectrum
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      theorem MaximalSpectrum.iInf_localization_eq_bot (R : Type u) [CommRing R] [IsDomain R] (K : Type v) [Field K] [Algebra R K] [IsFractionRing R K] :
      ⨅ (v : MaximalSpectrum R), Localization.subalgebra.ofField K v.asIdeal.primeCompl ⋯ = ⊥

      An integral domain is equal to the intersection of its localizations at all its maximal ideals viewed as subalgebras of its field of fractions.

      source
      theorem PrimeSpectrum.iInf_localization_eq_bot (R : Type u) [CommRing R] [IsDomain R] (K : Type v) [Field K] [Algebra R K] [IsFractionRing R K] :
      ⨅ (v : PrimeSpectrum R), Localization.subalgebra.ofField K v.asIdeal.primeCompl ⋯ = ⊥

      An integral domain is equal to the intersection of its localizations at all its prime ideals viewed as subalgebras of its field of fractions.