Commutative Algebra/Spectrum with Zariski topology

From Wikibooks, open books for an open world
Jump to navigation Jump to search

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[edit]

Definition 16.2:

Let be a ring and a subset of . Then define

.

The sets , where ranges over subsets of , satisfy the following equations:

Proposition 16.3:

Let be a ring, and let be a family of subsets of .

  1. and
  2. 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.

Definition 16.4:

Principal open sets[edit]

Topological properties of the spectrum[edit]