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Zariski Topology

Definitions.

(a)
An ideal of a ring is called prime if whenever for , then or (or both).

(b)
A variety is called irreducible if for any decomposition of into a union of two subvarieties, either or . In other words, there are no non-trivial decompositions of into smaller varieties.

Proposition 2   A variety is irreducible iff its ideal is prime.

Thus, there exists a one-to-one correspondence between the set of varieties and the set of prime (homogeneous if projective ) ideals in the corresponding polynomial ring.

Theorem 4   Any radical ideal is uniquely expressible as a finite intersection of prime ideals with for . Equivalently, any variety can be uniquely expressed as a finite union of irreducible subvarieties with for .

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

Definition. Let be a set of points in some space. A topology on is a set of designated subsets of , called the open sets of , 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)
and are open.

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

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

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

where ranges over polynomials (homogeneous if projective ).

Lemma 2   The Zariski topology is indeed a topology on .

Exercise. Show that the Zariski topology on the projective line is different from the analytic topology of .

Many statements in algebraic geometry are true for general points on varieties, i.e. if is a variety and is an open dense set of , then any point is called a general point on . (If 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 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.)

Next: Bezout's Theorem Up: Algebraic Geometry Previous: Examples of Varieties
Zvezdelina Stankova-Frenkel 2001-01-24