Zariski topology

Definition (Zariski topology on Spec(R))

Let $R$ be a commutative ring and let $X=\operatorname{Spec}(R)=Spec(R)$.

For an ideal $I\subseteq R$, define

The Zariski closed sets are precisely the subsets $V(I)$ as $I$ ranges over ideals of $R$. This defines a topology on $X$, called the Zariski topology.

Basic open sets

For $f\in R$, define the basic open set

The sets $D(f)$ form a basis for the Zariski topology.

Key identities

For ideals $I,J\subseteq R$ and elements $f,g\in R$:

• $V(0)=X$, and $V(R)=\varnothing$.
• $V(I)\cap V(J)=V(I+J)$.
• $V(I)\cup V(J)=V(IJ)$.
• $D(f)\cap D(g)=D(fg)$.

Examples

1. Spec($\mathbb Z$).
   In $X=\operatorname{Spec}(\mathbb Z)$, the closed set $V((n))$ consists of the primes $(p)$ with $p\mid n$ (and $V((0))=X$).
   The basic open set $D(n)$ is the set of prime ideals not containing $n$, i.e. primes $(p)$ with $p\nmid n$, together with $(0)$.

2. Spec($k[x]$) and the principal open $D(f)$.
   For a field $k$, $D(f)\subset \operatorname{Spec}(k[x])$ consists of primes $\mathfrak p$ such that $f\notin\mathfrak p$.
   Over an algebraically closed field, this corresponds to removing the (finite) set of closed points where $f$ vanishes.

3. Induced topology on MaxSpec.
   The maximal spectrum $\operatorname{MaxSpec}(R)=MaxSpec(R)$ inherits the subspace topology from $X=\operatorname{Spec}(R)$. Its closed sets are $V(I)\cap \operatorname{MaxSpec}(R)$.

Related knowls

• Spectra: prime spectrum , maximal spectrum
• Ideals: ideal , prime ideal
• Localization viewpoint: localization , localization at a prime