proof that Spec(R) is quasi-compact

Note that most of the notation used here is defined in the entry prime spectrum.

The following is a proof that Spec(R) is quasi-compact.


Let Λ be an indexing set and {Uλ}λΛ be an open cover for Spec(R). For every λΛ, let Iλ be an ideal of R with Uλ=Spec(R)V(Iλ). Since


V(λΛIλ)=. Thus, by this theorem (, λΛIλ=R. Since 1RR=λΛIλ, there exists a finite subset L of Λ such that, for every L, there exists an iI with 1R=Li.

Let rR. Then r=r1R=rLi=LriLI. Thus, LI=R. Therefore, V(LI)=. Since


{Uλ}λΛ restricts to a finite subcover. It follows that Spec(R) is quasi-compact. ∎

Title proof that Spec(R) is quasi-compact
Canonical name ProofThatoperatornameSpecRIsQuasicompact
Date of creation 2013-03-22 16:07:40
Last modified on 2013-03-22 16:07:40
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 10
Author Wkbj79 (1863)
Entry type Proof
Classification msc 14A15
Related topic VIemptysetImpliesIR