Theorems · Theorem · algebraic geometry

PrimeSpectrum.isClosed_range_of_stableUnderSpecialization

∀ {R : Type u_1} {S : Type u_2} [inst : CommSemiring R] [inst_1 : CommSemiring S] (f : R →+* S),
  StableUnderSpecialization (Set.range (PrimeSpectrum.comap f)) → IsClosed (Set.range (PrimeSpectrum.comap f))

[Stacks Tag 00HY](https://stacks.math.columbia.edu/tag/00HY)

Defined in: Mathlib.RingTheory.Spectrum.Prime.Topology
Cited by: 2 results in Mathlib
Foundations: Depth 84 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CommSemiring CommSemiring

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Cites12

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

Setproof · cited by 53,352
CommSemiringstatement and proof · cited by 10,911
RingHomstatement and proof · cited by 10,189
Set.rangestatement and proof · cited by 4,705
Set.univproof · cited by 3,945
IsClosedstatement and proof · cited by 1,639
PrimeSpectrumstatement and proof · cited by 625
Set.image_univproof · cited by 322
PrimeSpectrum.comapstatement and proof · cited by 199
isClosed_univproof · cited by 43
StableUnderSpecializationstatement and proof · cited by 32
PrimeSpectrum.isClosed_image_of_stableUnderSpecializationproof · cited by 6

Cited by2

Results whose statement or proof uses this declaration.

PrimeSpectrum.isClosed_of_stableUnderSpecialization_of_isConstructibleproof · cited by 1
PrimeSpectrum.stableUnderSpecialization_range_iffproof · cited by 0