411 REFERENCE · Wikibooks
Commutative Algebra/Spectrum with Zariski topology
Reference material by Wikibooks contributors. This retains its source identity and is not 411 news reporting.
0 100% #FFFFFF Definition 16.1:
Let be a commutative ring. The spectrum of is the set
;
i.e. the set of all prime ideals of .
On , we will define a topology, turning into a topological space. This topology will be called Zariski topology, although only Alexander Grothendieck gave the definition in the above generality.
Closed sets
0 100% #FFFFFF Definition 16.2:
Let be a ring and a subset of . Then define
.
The sets , where ranges over subsets of , satisfy the following equations:
0 100% #FFFFFF Proposition 16.3:
Let be a ring, and let be a family of subsets of .
and
If is finite, then .
Proof:
The first two items are straightforward. For the third, we use induction on . is clear; otherwise, the direction is clear, and the other direction follows from lemma 14.20.
0 100% #FFFFFF Definition 16.4:
Principal open sets
Topological properties of the spectrum