Zariski Topology

Definitions.

(a)

An ideal $P$ of a ring $R$ is called prime if whenever $ab\in P$ for $a,b\in R$, then $a\in P$ or $b\in P$ (or both).

(b)

A variety $X$ is called irreducible if for any decomposition $X=X_1\cup X_2$ of $X$ into a union of two subvarieties, either $X_1=X$ or $X_2=X$. In other words, there are no non-trivial decompositions of $X$ into smaller varieties.

Proposition 2   A variety $X$ is irreducible iff its ideal $I(X)$ is prime.

Thus, there exists a one-to-one correspondence between the set of varieties $\mathcal{X}$ and the set of prime (homogeneous if projective $X$) ideals $\mathcal{P}$ in the corresponding polynomial ring.

Theorem 4   Any radical ideal $I\subset K[x_1,...,x_n]$ is uniquely expressible as a finite intersection of prime ideals $P_i$ with $P_i\not\subset P_j$ for $i\not = j$. Equivalently, any variety $X$ can be uniquely expressed as a finite union of irreducible subvarieties $X_i$ with $X_i\not\subset X_j$ for $i\not = j$.

The varieties $X_i$ appearing in this unique decomposition are called the irreducible components of $X$.

Definition. Let $X$ be a set of points in some space. A topology on $X$ is a set $\mathcal{T}$ of designated subsets of $X$, called the open sets of $X$, so that the following axioms are satisfied:

(a)

The union of any collection of open sets is open.

(b)

The intersection of any finite collection of open sets is open.

(c)

$X$ and $\emptyset$ are open.

The closed sets in $X$ are the complements of the open sets.

We define below the so-called Zariski topology on algebraic varieties. If we work over $\mathbb{C}$, every variety can be roughly viewed as a complex manifold $X$ (with the exception of a proper subset of its singular points). Through its embedding in, say, $\mathbb{C}^n$, $X$ will inherit the usual complex analytic topology from $\mathbb{C}^n$ - a basis for the open sets on $X$ will consist of the intersections of $X$ with any finite balls in $\mathbb{C}^n$.

The Zariski topology is a different kind of topology. A basis for the open sets in $X$ is given by the sets

$$U_f=\{p\in X\,\,\vert\,\,f(p)\not = 0\}$$

where $f$ ranges over polynomials (homogeneous if projective $X$).

Lemma 2   The Zariski topology is indeed a topology on $X$.

Exercise. Show that the Zariski topology on the projective line $\mathbb{P}^1_{\mathbb{C}}$ is different from the analytic topology of $\mathbb{P}^1_{\mathbb{C}}$.

Many statements in algebraic geometry are true for general points on varieties, i.e. if $X$ is a variety and $U$ is an open dense set of $X$, then any point $p\in U$ is called a general point on $U$. (If $X$ is irreducible, then any nonempty open set will be dense. This, in particular, makes Zariski topology a non-Housdorff topology - in the latter, one needs for any two points of $X$ to have two nonintersecting open sets containing each one of the points. This confirms once again that the Zariski topology is much coarser than the analytic topology.)