Theorems · Definition · algebraic geometry

Ideal.minimalPrimes.equivIrreducibleComponents

{R : Type u} →
  [inst : CommSemiring R] →
    (I : Ideal R) → ↑I.minimalPrimes ≃o (↑(irreducibleComponents ↑(PrimeSpectrum.zeroLocus ↑I)))ᵒᵈ

Zero loci of minimal prime ideals over I are irreducible components in zeroLocus I and any irreducible component is a zero locus of some minimal prime ideal.

Defined in: Mathlib.RingTheory.Spectrum.Prime.Topology
Cited by: 0 results in Mathlib
Foundations: Depth 91 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CommSemiring

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Setstatement · cited by 53,352
CommSemiringstatement and proof · cited by 10,911
SetLike.coestatement and proof · cited by 8,199
Set.Elemstatement and proof · cited by 7,166
Set.ofPredproof · cited by 6,101
Idealstatement and proof · cited by 4,748
OrderDualstatement · cited by 927
OrderIsostatement and proof · cited by 874
Ideal.IsPrimeproof · cited by 827
PrimeSpectrumstatement · cited by 625
PrimeSpectrum.asIdealproof · cited by 333
PrimeSpectrum.zeroLocusstatement and proof · cited by 164

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