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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
![$ I\subset
K[x_1,...,x_n]$](img12.gif)
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