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# Varieties, Ideals, Nullstellensatz

Let be a field. We shall work over , meaning, our coefficients of polynomials and other scalars will lie in .

Definitions.

1.
An affine variety in is the zero locus of a collection of polynomials in . A projective variety X in is the zero locus of a collection of homogeneous polynomials in .

2.
For a given variety , the set of all polynomials vanishing on is an ideal, called the ideal of and denoted by . In other words,

or

depending on whether the variety is affine or projective. In the second case, the ideal is called homogeneous, i.e. generated by homogeneous polynomials. Conversely, for a given ideal (or homogeneous ), the zero locus of is denoted by .

Projective varieties can be thought of as completions'', compactifications'', or closures'' of affine varieties. Their global properties are usually easier to describe than those of affine varieties. Conversely, affine varieties can be thought of as building blocks of projective varieties (indeed, they constitute an open cover), and hence local properties are easier to describe using affine varieties. However, projective varieties vary nicely'' in families and hence parametrizing and moduli spaces are usually constructed for projective varieties with certain defining common properties.

3.
A ring is called Noetherian if any inscreasing sequence of ideals terminates, i.e. whenever 's are ideals in such that

then for some :

4.
An ideal in a ring is called radical if whenever (, ), then . In other words, contains all (positive integer) roots of its elements.

Theorem 1   The polynomial ring is Noetherian. Consequently, any ideal of is finitely generated. In particular, for any affine (or projective) variety , the ideal of is generated by finitely many polynomials.

Lemma 1   The ideal of any algebraic variety is radical. If is an arbitrary ideal, the set of all radicals of its elements:

for some

is also an ideal, called the radical of . If is a radical ideal, then its radical is itself, i.e. the operation of taking radicals stabilizes after one step.

We construct maps bewteen the set of all varieties over and the set of all ideals by sending

and

It is immediate from definition of an ideal of variety that , i.e. id. Also, the image of is inside , the subset of radical ideals in . Thus, we have an injection with a one-way inverse . It is natural to ask whether and are inverses of each other, i.e. whether id.

For any ideal (not necessarily radical), we consider - a variety, and then take . It is evident that will be a radical ideal containing , but is it going to be ? To paraphrase the problem, start with being a radical ideal and take . Is this equal to ?

The answer in general is no. For example, if is the ground field, and is the ideal generated by the single polynomial in the affine plane, then is obviously radical ( is irreducible), and the zero locus of is - just one point. However, the ideal of is definitely much larger than - it consists of all polynomials vanishing at , i.e. having no free terms: . Thus, we end up with a (radical) ideal bigger than the original.

The above situation is possible because is not an algebraically closed field. This leads to the famous Nullstelensatz, a basic theorem in commutative algebra, on which much of algebraic geometry over algebraically closed fields is based.

Theorem 2 (Nullstelensatz)   If is an algebraically closed field, then for any ideal :

In particular, there is a one-to-one correspondence between the set of affine varieties in and radical ideals given by

and

Note that the radical of the unit ideal is again the unit ideal: . This implies the following corollary:

Corollary 1   If are polynomials in several variables over an algebraically closed field , then they have no common zeros in iff

for some polynomials .

Some other strange'' things happen over fields, which are not algebraically closed. For example, we would like to call a planar curve'' any variety in which is given by 1 polynomial. However, over , the curve'' defined by is really just a point, while over (or any algebraically closed fields) it is a pair of intersecting lines. Thus, many interesting and intuitive properties of algebraic varieties hold only over algebraically closed fields.

There is an analog of Nullstelensatz for projective varieties (for , of course.) There is one subtle point, though. We call the irrelevant ideal in . Note that is radical, and that . Yet, is not the whole ideal of : , the unit ideal, is the ideal of . Thus, we have two radical ideals competing for the : . The bigger one wins'', because , and we state the Nullstelensatz as follows:

Theorem 3   There is a one-to-one correspondence between the set of projective varieties and the set of radical homogeneous ideals minus given by and from above. In particular, for :

Note further that for a (homogeneous) ideal , iff or . In both cases, , which can be shown to imply for some .

Proposition 1   Let be a homogeneous ideal. Then iff contains a power of the irrelevant ideal. In other words, a collection of homogeneous polynomials will have no common zeros iff the ideal generated by the 's contains all (homogeneous) polynomials of a certain degree .

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