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.^[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.

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.^[1] 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 (irreducible) algebraic varieties, which correspond to prime ideals. Thus the Zariski topology on the set of prime ideals (spectrum) 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.

V(S) = V((S)), where (S) is the generated by the elements of S;

ideal

For any two ideals of polynomials I, J, we have
$V(I)\cup V(J)\,=\,V(IJ);$

$V(I)\cap V(J)\,=\,V(I+J).$

Spec k, the spectrum of a k is the topological space with one element.

field

Spec $\mathbb {Z}$ , the spectrum of the has a closed point for every prime number p corresponding to the maximal ideal $(p)\subseteq \mathbb {Z}$ , and one non-closed generic point (i.e., whose closure is the whole space) corresponding to the zero ideal (0). So the closed subsets of Spec $\mathbb {Z}$ are precisely the whole space and the finite unions of closed points.

integers

Spec k[t], the spectrum of the over a field k: such a polynomial ring is known to be a principal ideal domain and the irreducible polynomials are the prime elements of k[t]. If k is algebraically closed, for example the field of complex numbers, a non-constant polynomial is irreducible if and only if it is linear, of the form t − a, for some element a of k. So, the spectrum consists of one closed point for every element a of k and a generic point, corresponding to the zero ideal, and the set of the closed points is homeomorphic with the affine line k equipped with its Zariski topology. Because of this homeomorphism, some authors use the term affine line for the spectrum of k[t]. If k is not algebraically closed, for example the field of the real numbers, the picture becomes more complicated because of the existence of non-linear irreducible polynomials. In this case, the spectrum consists of one closed point for each monic irreducible polynomial, and a generic point corresponding to the zero ideal. For example, the spectrum of $\mathbb {R} [t]$ consists of the closed points (x − a), for a in $\mathbb {R}$ , the closed points (x² + px + q) where p, q are in $\mathbb {R}$ and with negative discriminant p² − 4q < 0, and finally a generic point (0). For any field, the closed subsets of Spec k[t] are finite unions of closed points, and the whole space. (This results from the fact that k[t] is a principal ideal domain, and, in a principal ideal domain, the prime ideals that contain an ideal are the prime factors of the prime factorization of a generator of the ideal).

Zariski topology

ideal

field

integers

polynomial ring

Spectral space