Zariski topology

In algebraic geometry and commutative algebra, the Zariski topology is a topology defined on geometric objects called varieties. It is very different from topologies that are commonly used in real or complex analysis; in particular, it is not Hausdorff. This topology was introduced primarily by Oscar Zariski and later generalized for making the set of prime ideals of a commutative ring a topological space.
The Zariski topology allows tools from topology to be used to study algebraic varieties, even when the underlying field is not a topological field. This is one of the basic ideas of scheme theory, which allows one to build general algebraic varieties by gluing together affine varieties in a way similar to that in manifold theory, where manifolds are built by gluing together charts, which are open subsets of real affine spaces.
The Zariski topology of an algebraic variety is the topology whose closed sets are the algebraic subsets of the variety. In the case of an algebraic variety over the complex numbers, the Zariski topology is thus coarser than the usual topology, as every algebraic set is closed for the usual topology.
The generalization of the Zariski topology to the set of prime ideals of a commutative ring follows from Hilbert's Nullstellensatz, that establishes a bijective correspondence between the points of an affine variety defined over an algebraically closed field and the maximal ideals of the ring of its regular functions. This suggests defining the Zariski topology on the set of the maximal ideals of a commutative ring as the topology such that a set of maximal ideals is closed if and only if it is the set of all maximal ideals that contain a given ideal. Another basic idea of Grothendieck's scheme theory is to consider as points, not only the usual points corresponding to maximal ideals, but also all algebraic varieties, which correspond to prime ideals. Thus the Zariski topology on the set of prime ideals of a commutative ring is the topology such that a set of prime ideals is closed if and only if it is the set of all prime ideals that contain a fixed ideal.

Zariski topology of varieties

In classical algebraic geometry, the Zariski topology is defined on algebraic varieties. The Zariski topology, defined on the points of the variety, is the topology such that the closed sets are the algebraic subsets of the variety. As the most elementary algebraic varieties are affine and projective varieties, it is useful to make this definition more explicit in both cases. We assume that we are working over a fixed, algebraically closed field k.

Affine varieties

First, we define the topology on the affine space formed by the -tuples of elements of. The topology is defined by specifying its closed sets, rather than its open sets, and these are taken simply to be all the algebraic sets in That is, the closed sets are those of the form
where S is any set of polynomials in n variables over k. It is a straightforward verification to show that:

V = V), where is the ideal generated by the elements of S;
For any two ideals of polynomials I, J, we have
#
#

It follows that finite unions and arbitrary intersections of the sets V are also of this form, so that these sets form the closed sets of a topology. Writing, the sets are open sets, known as the principal ''open sets, that form a base for the topology. This is the Zariski topology on
If X'' is an affine algebraic set then the Zariski topology on it is defined simply to be the subspace topology induced by its inclusion into some Equivalently, it can be checked that:

The elements of the affine coordinate ring act as functions on X just as the elements of act as functions on ; here, I is the ideal of all polynomials vanishing on X.
For any set of polynomials S, let T be the set of their images in A. Then the subset of X is equal to the intersection with X of V.

This establishes that the above equation, clearly a generalization of the definition of the closed sets in above, defines the Zariski topology on any affine variety.

Projective varieties

Recall that -dimensional projective space is defined to be the set of equivalence classes of non-zero points in by identifying two points that differ by a scalar multiple in. The elements of the polynomial ring are not generally functions on because any point has many representatives that yield different values in a polynomial; however, for homogeneous polynomials the condition of having zero or nonzero value on any given projective point is well-defined since the scalar multiple factors out of the polynomial. Therefore, if is any set of homogeneous polynomials we may reasonably speak of
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, for sets of homogeneous polynomials, define a topology on As above the complements of these sets are denoted, or, if confusion is likely to result,.
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.

Properties

An important property of Zariski topologies is that they have a base consisting of simple elements, namely the for individual polynomials . That these form a base follows from the formula for the intersection of two Zariski-closed sets given above. The open sets in this base are called distinguished or basic open sets. The importance of this property results in particular from its use in the definition of an affine scheme.
By Hilbert's basis theorem and the fact that Noetherian rings are closed under quotients, every affine or projective coordinate ring is Noetherian. As a consequence, affine or projective spaces with the Zariski topology are Noetherian topological spaces, which implies that any closed subset of these spaces is compact.
However, except for finite algebraic sets, no algebraic set 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 is the zero set of the polynomials x₁ – a₁,..., x_n – a_n, 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 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

Spectrum of a ring

In modern algebraic geometry, an algebraic variety is often represented by its associated scheme, which is a topological space that is locally homeomorphic to the spectrum of a ring. The spectrum of a commutative ring ''A, denoted, is the set of the prime ideals of A'', equipped with the Zariski topology, for which the closed sets are the sets
where I is an ideal. Again writing, the distinguished open sets of are the sets for elements. These sets form a base for the Zariski topology.
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 are exactly the tuples such that the ideal generated by the polynomials 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, ' is "the same as" the maximal ideals containing S. To be slightly more explicit, let be the maximal spectrum of a commutative ring A, the set of its maximal ideals, and let k be an algebraically closed field. If I is an ideal, then under the classical definition of, there are one-to-one correspondences
given by
for, where are the images of under the natural projection. If I is a radical ideal, then is the coordinate ring of, which could be regarded as the ring of polynomial functions, the global regular functions on V.
Grothendieck's innovation in defining Spec for general commutative rings was to replace maximal ideals with all prime ideals; in this formulation, it is natural to simply generalize the correspondence between the closed set in affine space and the subset of the maximal spectrum of the polynomial ring that contains I to the definition of a closed set in the spectrum of a ring in the general commutative case.
In another approach, we first observe that we can reinterpret a polynomial as a function
mapping
where is the point to which corresponds via the weak Nullstellensatz. In other words, we map the ideal to f evaluated at. We can then regard an element of the polynomial ring as a function on the maximal ideals of the polynomial ring. Moreover, we observe that
or, generalizing to an ideal I,. We can thus view as the common set of "points" on which all the "functions" vanish.
Generalizing this approach to arbitrary commutative rings, another way to interpret the modern definition is to realize that the elements of A can actually be thought of as functions on its prime ideals,. Simply, any prime ideal has a corresponding residue field, which is the field of fractions of the quotient, and any element of A has an image in this residue field. Furthermore, the elements that are actually in are precisely those whose image vanishes at '. So if we think of the map, associated to any element of A:
, which assigns to each prime ideal the image of in the residue field of, as a function on , whose values, admittedly, may lie in different fields at different points, then we have
More generally, for any ideal I,, so is the common set of "points" on which all the "functions" vanish, which is formally analogous to the classical definition. In fact, as noted above, they agree in the sense that when A is the ring of polynomials over some algebraically closed field k, the maximal ideals of A are identified with n-tuples of elements of k, their residue fields are just k, and the "evaluation" maps are actually evaluation of polynomials at the corresponding n-tuples. For this special case, the classical definition is essentially the modern definition with only maximal ideals considered, and the interpretation of the modern definition as "zero sets of functions" agrees with the classical definition where they both make sense.
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.