Let be a field. We shall work over , meaning, our coefficients of polynomials and other scalars will lie in .
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.
We construct maps bewteen the set of all varieties over and the set of all ideals by sending
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.
Note that the radical of the unit ideal is again the unit ideal: . This implies the following corollary:
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:
Note further that for a (homogeneous) ideal , iff or . In both cases, , which can be shown to imply for some .