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In
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
and commutative algebra, the Zariski topology is a
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
which is primarily defined by its
closed set In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a clo ...
s. It is very different from topologies which are commonly used in the real or
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...
; in particular, it is not Hausdorff. This topology was introduced primarily by Oscar Zariski and later generalized for making the set of
prime ideal In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together wi ...
s of a
commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
(called the
spectrum A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors ...
of the ring) a topological space. The Zariski topology allows tools from
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
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 In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
theory, where manifolds are built by gluing together
charts A chart (sometimes known as a graph) is a graphical representation for data visualization, in which "the data is represented by symbols, such as bars in a bar chart, lines in a line chart, or slices in a pie chart". A chart can represent tab ...
, which are open subsets of real
affine space In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related ...
s. The Zariski topology of an algebraic variety is the topology whose
closed set In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a clo ...
s are the algebraic subsets of the variety. In the case of an algebraic variety over the
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s, 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 In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . Examples As an example, the field of real numbers is not algebraically closed, becaus ...
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.

# Zariski topology of varieties

In classical algebraic geometry (that is, the part of algebraic geometry in which one does not use schemes, which were introduced by Grothendieck around 1960), 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 set In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a clo ...
s 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 In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . Examples As an example, the field of real numbers is not algebraically closed, becaus ...
''k'' (in classical geometry ''k'' is almost always the
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s).

## Affine varieties

First, we define the topology on the
affine space In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related ...
$\mathbb^n,$ 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 $\mathbb^n.$ That is, the closed sets are those of the form $V(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 *# $V\left(I\right) \cup V\left(J\right)\,=\,V\left(IJ\right);$ *# $V\left(I\right) \cap V\left(J\right)\,=\,V\left(I + J\right).$ 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'') and called ''principal open sets'', form the topology itself). This is the Zariski topology on $\mathbb^n.$ If ''X'' is an affine algebraic set (irreducible or not) then the Zariski topology on it is defined simply to be the
subspace topology 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 topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced to ...
induced by its inclusion into some $\mathbb^n.$ Equivalently, it can be checked that: * The elements of the affine coordinate ring act as functions on ''X'' just as the elements of

## Projective varieties

Recall that ''n''-dimensional
projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
$\mathbb^n$ is defined to be the set of equivalence classes of non-zero points in $\mathbb^$ by identifying two points that differ by a scalar multiple in ''k''. The elements of the polynomial ring
homogeneous polynomial In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; ...
s 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 ''S'' is any set of homogeneous polynomials we may reasonably speak of :$V\left(S\right) = \.$ 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 $\mathbb^n.$ 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.

## Properties

An important property of Zariski topologies is that they have a base consisting of simple elements, namely the for individual polynomials (or for projective varieties, homogeneous polynomials) . 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 ). The open sets in this base are called ''distinguished'' or ''basic'' open sets. The importance of this property results in particular of its use in the definition of an affine scheme. By Hilbert's basis theorem and some elementary properties of Noetherian rings, every affine or projective coordinate ring is Noetherian. As a consequence, affine or projective spaces with the Zariski topology are
Noetherian topological space In mathematics, a Noetherian topological space, named for Emmy Noether, is a topological space in which closed subsets satisfy the descending chain condition. Equivalently, we could say that the open subsets satisfy the ascending chain condition ...
s, which implies that any closed subset of these spaces is
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Briti ...
. However, except for finite algebraic sets, no algebraic set is ever a
Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the many ...
. 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 $\mathbb^1.$

# Spectrum of a ring

In modern algebraic geometry, an algebraic variety is often represented by its associated scheme, which is a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called point ...
(equipped with additional structures) that is locally homeomorphic to the
spectrum of a ring In commutative algebra, the prime spectrum (or simply the spectrum) of a ring ''R'' is the set of all prime ideals of ''R'', and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with the ...
. 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 :$V\left(I\right) = \$ where ''I'' is an ideal. To see the connection with the classical picture, note that for any set ''S'' of polynomials (over an algebraically closed field), it follows from Hilbert's Nullstellensatz that the points of ''V''(''S'') (in the old sense) are exactly the tuples (''a''1, ..., ''an'') such that the ideal generated by the polynomials ''x''1 − ''a''1, ..., ''xn'' − ''an'' contains ''S''; moreover, these are maximal ideals and by the "weak" Nullstellensatz, an ideal of any affine coordinate ring is maximal if and only if it is of this form. Thus, ''V''(''S'') is "the same as" the maximal ideals containing ''S''. Grothendieck's innovation in defining Spec was to replace maximal ideals with all prime ideals; in this formulation it is natural to simply generalize this observation to the definition of a closed set in the spectrum of a ring. Another way, perhaps more similar to the original, to interpret the modern definition is to realize that the elements of ''A'' can actually be thought of as functions on the prime ideals of ''A''; namely, as functions on Spec ''A''. Simply, any prime ideal ''P'' has a corresponding residue field, which is the field of fractions of the quotient ''A''/''P'', and any element of ''A'' has a reflection in this residue field. Furthermore, the elements that are actually in ''P'' are precisely those whose reflection vanishes at ''P''. So if we think of the map, associated to any element ''a'' of ''A'': :$e_a \colon \bigl\left(P \in \operatornameA \bigr\right) \mapsto \left\left(\frac \in \operatorname\left(A/P\right)\right\right)$ ("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 :$e_a\left(P\right)=0 \Leftrightarrow P \in V\left(a\right)$ 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. Since as shown above, the classical definition is essentially the modern definition with only maximal ideals considered, this shows that 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 In algebraic geometry, Proj is a construction analogous to the spectrum-of-a-ring construction of affine schemes, which produces objects with the typical properties of projective spaces and projective varieties. The construction, while not funct ...
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.

## Examples

* Spec ''k'', the spectrum of a field ''k'' is the topological space with one element. * Spec ℤ, the spectrum of the
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language o ...
s has a closed point for every
prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
''p'' corresponding to the maximal ideal (''p'') ⊂ ℤ, and one non-closed
generic point In algebraic geometry, a generic point ''P'' of an algebraic variety ''X'' is, roughly speaking, a point at which all generic properties are true, a generic property being a property which is true for almost every point. In classical algebraic ...
(i.e., whose closure is the whole space) corresponding to the zero ideal (0). So the closed subsets of Spec ℤ are precisely the whole space and the finite unions of closed points. * Spec ''k'' 't'' the spectrum of the
polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variable ...
over a field ''k'': such a polynomial ring is known to be a
principal ideal domain In mathematics, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are princip ...
and the
irreducible polynomial In mathematics, an irreducible polynomial is, roughly speaking, a polynomial that cannot be factored into the product of two non-constant polynomials. The property of irreducibility depends on the nature of the coefficients that are accepted f ...
s are the
prime element In mathematics, specifically in abstract algebra, a prime element of a commutative ring is an object satisfying certain properties similar to the prime numbers in the integers and to irreducible polynomials. Care should be taken to distinguish prim ...
s of ''k'' 't'' If ''k'' is algebraically closed, for example the field of
complex numbers In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
, 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 call ''affine line'' the spectrum of ''k'' 't'' If ''k'' is not algebraically closed, for example the field of the
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s, the picture becomes more complicated because of the existence of non-linear irreducible polynomials. For example, the spectrum of ℝ 't''consists of the closed points (''x'' − ''a''), for ''a'' in ℝ, the closed points (''x''2 + ''px'' + ''q'') where ''p'', ''q'' are in ℝ and with negative discriminant ''p''2 − 4''q'' < 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 In commutative algebra, Krull's principal ideal theorem, named after Wolfgang Krull (1899–1971), gives a bound on the height of a principal ideal in a commutative Noetherian ring. The theorem is sometimes referred to by its German name, ''Krul ...
).

## Further 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 point In algebraic geometry, a generic point ''P'' of an algebraic variety ''X'' is, roughly speaking, a point at which all generic properties are true, a generic property being a property which is true for almost every point. In classical algebraic ...
s, which are the points with maximal closure, that is the minimal prime ideals. The closed points correspond to maximal ideals of ''A''. However, 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, say ''P'', does not contain the other. Then ''D''(''Q'') contains ''P'' but, of course, not ''Q''. 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 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.