Theorems · Theorem · commutative algebra

Algebra.QuasiFiniteAt.isClopen_singleton

∀ {R : Type u_1} {S : Type u_2} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] (p : PrimeSpectrum S)
  [IsArtinianRing R] [Algebra.FiniteType R S] [Algebra.QuasiFiniteAt R p.asIdeal], IsClopen {p}

If R is an artinian ring, and S is a finite type R-algebra R-quasi-finite at p, then {p} is clopen in Spec S.

Defined in

Mathlib.RingTheory.QuasiFinite.Basic

Cited by

2 results in Mathlib

Foundations

Depth 145 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingAlgebraIsArtinianRingAlgebra.FiniteTypeAlgebra.QuasiFiniteAt