In algebraic geometry and commutative algebra, the **Zariski topology** is a topology on algebraic varieties, introduced primarily by Oscar Zariski and later generalized for making the set of prime ideals of a commutative ring a topological space, called the spectrum of the ring.

- Zariski topology of varieties
- Affine varieties
- Projective varieties
- Properties
- Spectrum of a ring
- Examples
- Properties 2
- See also
- References
- Further reading

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 (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.

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.^{ [1] } 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* (in classical geometry *k* is almost always the complex numbers).

First we define the topology on the affine space formed by the n-tuples of elements of k. 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*(*S*) =*V*((*S*)), where (*S*) 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*(*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

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:

- The elements of the affine coordinate ring

act as functions on *X* just as the elements of act as functions on ; here, *I(X)* 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(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.

Recall that *n*-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 *k*. The elements of the polynomial ring are not 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 *S* 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 *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*).

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.

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.

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 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 (*a _{1}*, ...,

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 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.^{ [2] } The *spectrum of a commutative ring**A*, denoted Spec(*A*), 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.

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}*, ...,

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*:

("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. 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 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.

- 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
*p*corresponding to the maximal ideal (*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 the whole space and the finite unions of closed points. - Spec
*k*[*t*], the spectrum of the polynomial ring 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 call*affine line*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. 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).

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, 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 *T _{0}* spaces: given two points

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.

**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 problems about these sets of zeros.

In algebra and algebraic geometry, the **spectrum** of a commutative ring *R*, denoted by , is the set of all prime ideals of *R*. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space. A locally ringed space of this form is called an **affine scheme**.

In commutative algebra, the **Krull dimension** of a commutative ring *R*, named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally the Krull dimension can be defined for modules over possibly non-commutative rings as the deviation of the poset of submodules.

**Hilbert's Nullstellensatz** is a theorem that establishes a fundamental relationship between geometry and algebra. This relationship is the basis of algebraic geometry, a branch of mathematics. It relates algebraic sets to ideals in polynomial rings over algebraically closed fields. This relationship was discovered by David Hilbert who proved the Nullstellensatz and several other important related theorems named after him.

**Commutative algebra** is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative rings include polynomial rings; rings of algebraic integers, including the ordinary integers ; and *p*-adic integers.

**Algebraic varieties** are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Modern definitions generalize this concept in several different ways, while attempting to preserve the geometric intuition behind the original definition.

In commutative algebra and algebraic geometry, **localization** is a formal way to introduce the "denominators" to a given ring or module. That is, it introduces a new ring/module out of an existing one so that it consists of fractions such that the denominator *s* belongs to a given subset *S* of *R*. If *S* is the set of the non-zero elements of an integral domain, then the localization is the field of fractions: this case generalizes the construction of the ring **Q** of rational numbers from the ring **Z** of integers.

In mathematics and specifically in algebraic geometry, the **dimension** of an algebraic variety may be defined in various equivalent ways.

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 to parallelism and ratio of lengths for parallel line segments.

In algebraic geometry, an **affine variety**, or **affine algebraic variety**, over an algebraically closed field *k* is the zero-locus in the affine space *k*^{n} of some finite family of polynomials of *n* variables with coefficients in *k* that generate a prime ideal. If the condition of generating a prime ideal is removed, such a set is called an (affine) **algebraic set**. A Zariski open subvariety of an affine variety is called a quasi-affine variety.

In algebraic geometry, a **projective variety** over an algebraically closed field *k* is a subset of some projective *n*-space over *k* that is the zero-locus of some finite family of homogeneous polynomials of *n* + 1 variables with coefficients in *k*, that generate a prime ideal, the defining ideal of the variety. Equivalently, an algebraic variety is projective if it can be embedded as a Zariski closed subvariety of .

In mathematics, a **scheme** is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities and allowing "varieties" defined over any commutative ring.

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 functorial, is a fundamental tool in scheme theory.

In algebraic geometry, an **irreducible algebraic set** or **irreducible variety** is an algebraic set that cannot be written as the union of two proper algebraic subsets. An **irreducible component** is an algebraic subset that is irreducible and maximal for this property. For example, the set of solutions of the equation *xy* = 0 is not irreducible, and its irreducible components are the two lines of equations *x* = 0 and *y* =0.

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, since they are the complements of the closed subsets. The Noetherian property of a topological space can also be seen as a strong compactness condition, namely that every open subset of such a space is compact, and in fact it is equivalent to the seemingly stronger statement that *every* subset is compact.

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.

Affine geometry, broadly speaking, is the study of the geometrical properties of lines, planes, and their higher dimensional analogs, in which a notion of "parallel" is retained, but no metrical notions of distance or angle are. Affine spaces differ from linear spaces in that they do not have a distinguished choice of origin. So, in the words of Marcel Berger, "An affine space is nothing more than a vector space whose origin we try to forget about, by adding translations to the linear maps." Accordingly, a **complex affine space**, that is an affine space over the complex numbers, is like a complex vector space, but without a distinguished point to serve as the origin.

In mathematics, a proper ideal of a commutative ring is said to be **irreducible** if it cannot be written as the intersection of two strictly larger ideals.

This is a **glossary of algebraic geometry**.

In algebraic geometry, the **main theorem of elimination theory** states that every projective scheme is proper. A version of this theorem predates the existence of scheme theory. It can be stated, proved, and applied in the following more classical setting. Let *k* be a field, denote by the *n*-dimensional projective space over *k*. The main theorem of elimination theory is the statement that for any *n* and any algebraic variety V defined over *k*, the projection map sends Zariski-closed subsets to Zariski-closed subsets.

- ↑ Mumford, David (1999) [1967],
*The red book of varieties and schemes*, Lecture Notes in Mathematics,**1358**(expanded, Includes Michigan Lectures (1974) on Curves and their Jacobians ed.), Berlin, New York: Springer-Verlag, doi:10.1007/b62130, ISBN 978-3-540-63293-1, MR 1748380 - ↑ Dummit, D. S.; Foote, R. (2004).
*Abstract Algebra*(3 ed.). Wiley. pp. 71–72. ISBN 9780471433347.

- Hartshorne, Robin (1977),
*Algebraic Geometry*, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157, OCLC 13348052 - Todd Rowland. "Zariski Topology".
*MathWorld*.

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