Zariski topology
It is very different from topologies that are commonly used in real or complex analysis; in particular, it is not Hausdorff.
[1] This topology was introduced primarily by Oscar Zariski and later generalized for making the set of prime ideals of a commutative ring (called the spectrum of the ring) a topological space.
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 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.
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 (irreducible) algebraic varieties, which correspond to prime ideals.
Equivalently, it can be checked that: This establishes that the above equation, clearly a generalization of the definition of the closed sets in
by identifying two points that differ by a scalar multiple in k. The elements of the polynomial ring
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.
An important property of Zariski topologies is that they have a base consisting of simple elements, namely the D(f) for individual polynomials (or for projective varieties, homogeneous polynomials) f. 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)).
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.
Every regular map of varieties is continuous in the Zariski topology.
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
In modern algebraic geometry, an algebraic variety is often represented by its associated scheme, which is a topological space (equipped with additional structures) that is locally homeomorphic to the spectrum of a ring.
To see the connection with the classical picture, note that for any set S of polynomials (over an algebraically closed field), it follows from Hilbert's Nullstellensatz that the points of V(S) (in the old sense) are exactly the tuples (a1, ..., an) such that the ideal generated by the polynomials 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.
Furthermore, the elements that are actually in P are precisely those whose reflection vanishes at P. So if we think of the map, associated to any element a of A: ("evaluation of a"), which assigns to each point its reflection in the residue field there, as a function on Spec A (whose values, admittedly, lie in different fields at different points), then we have More generally, V(I) for any ideal I is the common set on which all the "functions" in I vanish, which is formally similar to the classical definition.
In fact, 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 (as discussed in the previous paragraph) 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.
Just as in classical algebraic geometry, any spectrum or projective spectrum is (quasi)compact, and if the ring in question is Noetherian then the space is a Noetherian topological 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.
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.
