Zariski topology

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In mathematics, namely algebraic geometry, the Zariski topology is a particular topology chosen for algebraic varieties that reflects the algebraic nature of their definition but is only weakly related to their geometric properties; it is due to Oscar Zariski and took a place of particular importance in the field around 1950. Joe Harris likes to say in his introductory lectures that it is "not a real topology" and points out that in the Zariski topology, every two algebraic curves are homeomorphic simply because their underlying sets have equal cardinalities and their topologies are both cofinite. Naturally, such a homeomorphism is not a regular map (see algebraic geometry#Regular functions), but this merely highlights the fact that the deep structure of algebraic varieties is mostly encoded in the choice of functions between them rather than of topologies on them. In this sense, the Zariski topology is an organizational tool rather than an object of study (compare with the role of the topology in algebraic topology). The more subtle étale topology was discovered by Grothendieck in the 1960s; while it reflects the geometry far more accurately it is also highly nontrivial even to describe and is not as basic to the subject.

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[edit] The classical definition

In classical algebraic geometry (that is, the subject prior to the Grothendieck revolution of the late 1950s and 1960s) the Zariski topology was defined in the following way. Just as the subject itself was divided into the study of affine and projective varieties (see algebraic variety#Formal definitions) the Zariski topology is defined slightly differently for these two. We assume that we are working over a fixed, algebraically closed field k, which in classical geometry was almost always the complex numbers.

[edit] Affine varieties

First we define the topology on affine spaces \mathbb{A}^n, which as sets are just n-dimensional vector spaces over k. The topology is defined by specifying its closed, rather than its open sets, and these are taken simply to be all the algebraic sets in \mathbb{A}^n. That is, the closed sets are those of the form

V(S) = \{x \in \mathbb{A}^n \mid f(x) = 0, \forall f \in S\}

where S is any set of polynomials in n variables over k. It is a straightforward verification to show that:

  • V(S) = V((S)), where (S) is the ideal generated by the elements of S;
  • For any two ideals of polynomials I, J, we have
    1. V(I) \cup V(J)\,=\,V(IJ);
    2. V(I) \cap V(J)\,=\,V(I + J).

It follows that finite unions and arbitrary intersections of the sets V(S) are also of this form, so that these sets form the closed sets of a topology (equivalently, their complements, denoted D(S), form the topology itself). This is the Zariski topology on \mathbb{A}^n.

If X is an affine algebraic set (irreducible or not) then the Zariski topology on it is defined simply to be the subspace topology induced by its inclusion into some \mathbb{A}^n. Equivalently, it can be checked that:

  • The elements of the affine coordinate ring
A(X)\,=\,k[x_1, \dots, x_n]/I(X)

act as functions on X just as the elements of k[x_1, \dots, x_n] act as functions on \mathbb{A}^n;

  • For any set of polynomials S, let T be the set of their images in A(X). Then the subset of X
V'(T) = \{x \in X \mid f(x) = 0, \forall f \in T\}

(these notations are not standard) is equal to the intersection with X of V(S).

This establishes that the above equation, clearly a generalization of the previous one, defines the Zariski topology on any affine variety.

[edit] Projective varieties

Recall that n-dimensional projective space \mathbb{P}^n is defined to be the set of equivalence classes of points in \mathbb{A}^{n + 1} by identifying two points which differ by a scalar multiple in k. The polynomial ring k[x_0, \dots, x_n] does not act as functions on \mathbb{P}^n because any point has many representatives which yield different values in a polynomial; however, the homogeneous polynomials do have well-defined zero or nonzero values on any projective point since the scalar multiple factors out of the polynomial. Therefore if S is any set of homogeneous polynomials we may reasonably speak of

V(S) = \{x \in \mathbb{P}^n \mid f(x) = 0, \forall f \in S\}.

The same facts as above may be established for these sets, except that the word "ideal" must be replaced by the phrase "homogeneous ideal", so that the V(S), for sets S of homogeneous polynomials, define a topology on \mathbb{P}^n. As above the complements of these sets are denoted D(S), or, if confusion is likely to result, D′(S).

The projective Zariski topology is defined for projective algebraic sets just as the affine one is defined for affine algebraic sets, by taking the subspace topology. Similarly, it may be shown that this topology is defined intrinsically by sets of elements of the projective coordinate ring, by the same formula as above.

[edit] Properties

A very useful fact about these topologies is that we may exhibit a basis for them consisting of particularly simple elements, namely the D(f) for individual polynomials (or for projective varieties, homogeneous polynomials) f. Indeed, that these form a basis follows from the formula for the intersection of two Zariski-closed sets given above (apply it repeatedly to the principal ideals generated by the generators of (S)). These are called distinguished or basic open sets.

Any variety, projective or affine, is a compact space with the Zariski topology. Indeed, more is true: by the Hilbert Basis Theorem and some elementary properties of Noetherian rings, every affine or projective coordinate ring is Noetherian. It follows from this that every open set is in fact a finite union of distinguished open sets, and it is easy to show that each distinguished open must be compact. As a consequence, every open set of every variety is compact, which makes them Noetherian topological spaces.

However, unless k is a finite field no variety is ever a Hausdorff space. In the old topological literature "compact" was taken to include the Hausdorff property, and this convention is still honored in algebraic geometry; therefore compactness in the modern sense is called "quasicompactness" in algebraic geometry. However, since every point (a1, ..., an) is the zero set of the polynomials x1 - a1, ..., xn - an, points are closed and so every variety satisfies the T1 axiom.

Every regular map of varieties is continuous in the Zariski topology. In fact, the Zariski topology is the weakest topology (with the fewest open sets) in which this is true and in which points are closed. This is easily verified by noting that the Zariski-closed sets are simply the intersections of the inverse images of 0 by the polynomial functions, considered as regular maps into \mathbb{A}^1.

[edit] The modern definition

Modern algebraic geometry takes the spectrum of a ring as its starting point. In this formulation, the Zariski-closed sets are taken to be the sets

V(I) = \{P \in \operatorname{Spec}\,(A) \mid I \subseteq P\}

where A is a fixed commutative ring and I is an ideal. To see the connection with the classical picture, note that for any set S of polynomials, it follows from Hilbert's Nullstellensatz that the points of V(S) are exactly the tuples (a1, ..., an) such that (x1 - a1, ..., xn - an) contains S; moreover, these are maximal ideals and by the "weak" Nullstellensatz, an ideal of any affine coordinate ring is maximal if and only if it is of this form. Thus, V(S) is "the same as" the maximal ideals containing S. Grothendieck's innovation in defining Spec was to replace maximal ideals with all prime ideals; in this formulation it is natural to simply generalize this observation to the definition of a closed set in the spectrum of a ring.

Just as Spec replaces affine varieties, the Proj construction replaces projective varieties in modern algebraic geometry. Just as in the classical case, to move from the affine to the projective definition we need only replace "ideal" by "homogeneous ideal", though there is a complication involving the "irrelevant maximal ideal" which is discussed in the cited article.

[edit] Properties

The most dramatic change in the topology from the classical picture to the new is that points are no longer necessarily closed; by expanding the definition, Grothendieck introduced generic points whose closures are strictly larger than themselves. The points which are closed are those which correspond to maximal ideals of A. Note, however, that the spectrum and projective spectrum are still T0 spaces: given two points P, Q, which are prime ideals of A, at least one of them does not contain the other, say P. Then D(Q) contains P but, of course, not Q.

Just as in classical algebraic geometry, any spectrum or projective spectrum is compact, and if the ring in question is Noetherian then the space is a Noetherian space. However, these facts are counterintuitive: we do not normally expect open sets, other than connected components, to be compact, and for affine varieties (for example, Euclidean space) we do not even expect the space itself to be compact. This is one instance of the geometric unsuitability of the Zariski topology. Grothendieck solved this problem by defining the notion of properness of a scheme (actually, of a morphism of schemes), which recovers the intuitive idea of compactness: Proj is proper, but Spec is not.

[edit] See also

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