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 (compared 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. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... A Möbius strip, a surface with only one side and one edge; such shapes are an object of study in topology. ... In classical algebraic geometry (and to some extent also in modern algebraic geometry), the main objects of study are algebraic varieties. ... Oscar Zariski was one of the most influential mathematicians working in the field of algebraic geometry in the twentieth century. ... Joseph Daniel Harris (born 1951), known nearly universally as Joe Harris, is a mathematician at Harvard University working in the field of algebraic geometry. ... In algebraic geometry, an algebraic curve is an algebraic variety of dimension equal to 1. ... This word should not be confused with homomorphism. ... In mathematics, a cofinite subset of a set X is a subset Y whose complement in X is a finite set. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... In mathematics, the étale cohomology theory of algebraic geometry is a refined construction of homological algebra, introduced in order to attack the Weil conjectures. ... Alexander Grothendieck (born March 28, 1928, Berlin) is one of the greatest mathematicians of the 20th century, with major contributions to algebraic geometry, homological algebra, and functional analysis. ...

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. In classical algebraic geometry (and to some extent also in modern algebraic geometry), the main objects of study are algebraic varieties. ... The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ...

Affine varieties

First we define the topology on affine spaces 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 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(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. 
  2. 

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 In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. ...

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 Equivalently, it can be checked that: In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a natural topology induced from that of X called the subspace topology (or the relative topology, or the induced topology). ...

• The elements of the affine coordinate ring

act as functions on X just as the elements of act as functions on

• For any set of polynomials S, let T be the set of their images in A(X). Then the subset of X

(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.

Projective varieties

Recall that n-dimensional projective space is defined to be the set of equivalence classes of points in by identifying two points which differ by a scalar multiple in k. The polynomial ring does not act as functions on 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 In mathematics, a projective space is a fundamental construction from any vector space. ... In mathematics, homogeneous has a variety of meanings. ...

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 As above the complements of these sets are denoted D(S), or, if confusion is likely to result, D′(S). In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field (or commutative ring) with an extra piece of structure, known as a grading. ...

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

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. In mathematics, a base (or basis) B for a topological space X with topology T is a collection of open sets in T such that every open set in T can be written as a union of elements of B. We say that the base generates the topology T. Bases...

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. In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ... In mathematics, Hilberts basis theorem, first proved by David Hilbert in 1888, states that, if k is a field, then every ideal in the ring of multivariate polynomials k[x1, x2, ..., xn] is finitely generated. ... In abstract algebra, a Noetherian ring is a ring that satisfies the ascending chain condition on ideals. ... In mathematics, a Noetherian topological space is a topological space in which closed subsets satisfy the descending chain condition. ...

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, ..., a_n) is the zero set of the polynomials x₁ - a₁, ..., x_n - a_n, points are closed and so every variety satisfies the T₁ axiom. In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ... The title given to this article is incorrect due to technical limitations. ...

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 In topology and related areas of mathematics a continuous function is a morphism between topological spaces. ...

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 In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec(R), is defined to be the set of all proper prime ideals of R. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space. ...

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, ..., a_n) such that (x₁ - a₁, ..., x_n - a_n) 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. Hilberts Nullstellensatz (German: theorem of zeros) is a theorem in algebraic geometry that relates varieties and ideals in polynomial rings over algebraically closed fields. ...

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. Proj is a certain construction in mathematics, more precisely in the field of algebraic geometry. ...

Examples

The spectrum of ℤ

• Spec k, the spectrum of a field k is the topological space with one element.
• Spec ℤ, the spectrum of the integers has a closed point for every prime number corresponding to the maximal ideals (p) ⊂ 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 ℤ are precisely finite unions of closed points and the whole space.

• Spec k[t], spectrum of the polynomial ring over field k, which is also denoted , the affine line: the polynomial ring is known to be a principal ideal domain and the irreducible polynomials are the prime elements of the k[t]. If k is algebraically closed, e.g. the field of complex numbers, a non-constant polynomial is irreducible iff it is linear, i.e. 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. If k is not algebraically closed, for example the field of real numbers, the picture becomes more complicated because of the existence of non-linear irreducible polynomials. For example, the spectrum of consists of closed points (x-a), , (x²+px+q) where and 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 is clear from the above discussion for algebraically closed fields. The proof of the general case requires some commutative algebra, namely the fact, that the Krull dimension of k[t] is one -- see Krull's principal ideal theorem).

Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... The integers consist of the positive natural numbers (1, 2, 3, &#8230;) the negative natural numbers (&#8722;1, &#8722;2, &#8722;3, ...) and the number zero. ... In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. ... In mathematics, more specifically in ring theory a maximal ideal is an ideal which is maximal (with respect to set inclusion) amongst all proper ideals, i. ... In abstract algebra, a polynomial ring is the set of polynomials in one or more variables with coefficients in a ring. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, an affine space is an abstract structure that generalises the affine-geometric properties of Euclidean space. ... In abstract algebra, a principal ideal domain (PID) is an integral domain in which every ideal is principal (that is, generated by a single element). ... In mathematics, the adjective irreducible means that an object cannot be expressed as a product of at least two non-trivial factors in a given ring. ... In abstract algebra, an integral domain is a commutative ring with 0 &#8800; 1 in which the product of any two non-zero elements is always non-zero. ... In mathematics, a field F is said to be algebraically closed if every polynomial of degree at least 1, with coefficients in F, has a zero in F. In that case, every such polynomial splits into linear factors. ... The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ... IFF, Iff or iff can stand for: Interchange File Format - a computer file format introduced by Electronic Arts Identification, friend or foe - a radio based identification system utilizing transponders iff - the mathematics concept if and only if International Flavors and Fragrances - a company producing flavors and fragrances International Freedom Foundation... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In algebra, the discriminant of a polynomial is a certain expression in the coefficients of the polynomial which equals zero if and only if the polynomial has multiple roots in the complex numbers. ... In abstract algebra, commutative algebra studies commutative rings, their ideals, and modules based on such rings; and of fields and their algebras. ... In commutative algebra, the Krull dimension of a ring R, named after Wolfgang Krull (1899 - 1971), is defined to be the number of strict inclusions in a maximal chain of prime ideals. ... In commutative algebra, Krulls principal ideal theorem, named after Wolfgang Krull (1899 - 1971), gives a bound on the height of a principal ideal in a Noetherian ring. ...

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. In mathematics, in the fields of general topology and particularly of algebraic geometry, a generic point P of a topological space X is a point such that every point Q of X is a specialization of P, in the sense of the specialization order (or pre-order). ...

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. Connected and disconnected subspaces of R². The space A at top is connected; the shaded space B at bottom is not. ... In mathematics, a continuous function between topological spaces is called proper if inverse images of compact subsets are compact. ... In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. ...

See also

• Algebraic variety
• Spectrum of a ring
• Proj construction