Zariski topology - Citizendia

In mathematics, namely algebraic geometry, the Zariski topology is a particular topology chosen for algebraic varieties that reflects the algebraic nature of their definition. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of This article is about algebraic varieties For the term "a variety of algebras" and an explanation of the difference between a variety of algebras and an algebraic variety It is due to Oscar Zariski and took a place of particular importance in the field around 1950. Oscar Zariski (born Oscher Zaritsky 24 April 1899 in Kobrin, Poland (today Belarus) died 4 July Joe Harris says 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. 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 one Topological equivalence redirects here see also Topological equivalence (dynamical systems. In Mathematics, a cofinite Subset of a set X is a subset Y whose complement in X is a finite set Naturally, such a homeomorphism is not a regular map, 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. Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with 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). Algebraic topology is a branch of Mathematics which uses tools from Abstract algebra to study Topological spaces The basic goal is to find algebraic 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. In Category theory, a branch of Mathematics, a Grothendieck topology is a structure on a category C which makes the objects of C act like the Experimental infobox see Wikipedia talkPersondata before changing --> Alexander Grothendieck (born March 28, 1928 in Berlin, Germany

1 The classical definition
2 The modern definition
- 2.1 Examples
- 2.2 Properties
3 See also

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 the Algebraic variety definitions) the Zariski topology is defined slightly differently for these two. This article is about algebraic varieties For the term "a variety of algebras" and an explanation of the difference between a variety of algebras and an algebraic variety We assume that we are working over a fixed, algebraically closed field k, which in classical geometry was almost always the complex numbers. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted

Affine varieties

First we define the topology on affine spaces $\mathbb{A}^n,$ 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 $\mathbb{A}^n.$ That is, the closed sets are those of the form

$V(S) = \{x \in \mathbb{A}^n \mid f(x) = 0, \forall f \in S\}$

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
1. $V(I) \cup V(J)\,=\,V(IJ);$
2. $V(I) \cap V(J)\,=\,V(I + J).$

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). In Ring theory, a branch of Abstract algebra, an ideal is a special Subset of a ring. This is the Zariski topology on $\mathbb{A}^n.$

If X is an affine algebraic set (irreducible or not) then the Zariski topology on it is defined simply to be the subspace topology induced by its inclusion into some $\mathbb{A}^n.$ Equivalently, it can be checked that:

The elements of the affine coordinate ring

$A(X)\,=\,k[x_1, \dots, x_n]/I(X)$

act as functions on X just as the elements of $k[x_1, \dots, x_n]$ act as functions on $\mathbb{A}^n;$

For any set of polynomials S, let T be the set of their images in A(X). In Topology and related areas of Mathematics, a subspace of a Topological space X is a Subset S of X which is Then the subset of X

$V'(T) = \{x \in X \mid f(x) = 0, \forall f \in T\}$

(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 $\mathbb{P}^n$ is defined to be the set of equivalence classes of non-zero points in $\mathbb{A}^{n + 1}$ by identifying two points which differ by a scalar multiple in k. In Mathematics a projective space is a set of elements constructed from a vector space such that a distinct element of the projective space consists of all non-zero vectors which The polynomial ring $k[x_0, \dots, x_n]$ does not act as functions on $\mathbb{P}^n$ 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. In Mathematics, a homogeneous polynomial is a Polynomial whose terms are Monomials all having the same total degree; or are elements of the same Therefore if S is any set of homogeneous polynomials we may reasonably speak of

$V(S) = \{x \in \mathbb{P}^n \mid f(x) = 0, \forall f \in S\}.$

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 $\mathbb{P}^n.$ 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

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. In Mathematics, a base (or basis) B for a Topological space X with topology T is a collection of Open sets 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.

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. In Mathematics, Hilbert's basis theorem states that every ideal in the ring of multivariate polynomials over a field is finitely generated In Abstract algebra, a Noetherian ring is a ring that satisfies the Ascending chain condition on ideals. 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 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 Topology and related branches of Mathematics, a Hausdorff space, separated space or T2 space is a Topological 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, T1 spaces and R0 spaces are particular kinds of Topological spaces The

Every regular map of varieties is continuous in the Zariski topology. In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function 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 $\mathbb{A}^1.$

The modern definition

Modern algebraic geometry takes the spectrum of a ring as its starting point. In Abstract algebra and Algebraic geometry, the spectrum of a Commutative ring R, denoted by Spec( R) is defined to be the set of In this formulation, the Zariski-closed sets are taken to be the sets

$V(I) = \{P \in \operatorname{Spec}\,(A) \mid I \subseteq P\}$

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 (over an algebraically closed field), it follows from Hilbert's Nullstellensatz that the points of V(S) are exactly the tuples (a₁, . Hilbert's Nullstellensatz ( German: "theorem of zeros" is a theorem in Algebraic geometry, a branch of Mathematics, that relates Algebraic . . , 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.

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. In Mathematics, the residue field is a basic construction in Commutative algebra. In Mathematics, every Integral domain can be embedded in a field; the smallest field which can be used is the field of fractions or field of quotients Furthermore, the elements which are actually in P are precisely those whose reflection vanishes. So if we think of the map, associated to any element a of A:

$e_a \colon \bigl(P \in \operatorname{Spec}(A)\bigr) \mapsto \left(\frac{a \bmod P}{1} \in \operatorname{Frac}(A/P)\right)$

("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 this function vanishes precisely at the points of V((a)). 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. In Algebraic geometry, Proj is a construction analogous to the spectrum-of-a-ring construction of Affine schemes which produces objects with the typical 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

The spectrum of ℤ

Spec k, the spectrum of a field k is the topological space with one element. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division
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. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 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 In Mathematics, in the fields of General topology and particularly of Algebraic geometry, a generic point P of a Topological space 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], the spectrum of the polynomial ring over a field k, which is also denoted $\mathbb{A}^1$ , the affine line: the polynomial ring is known to be a principal ideal domain and the irreducible polynomials are the prime elements of k[t]. In Mathematics, especially in the field of Abstract algebra, a polynomial ring is a ring formed from the set of Polynomials in one or more variables In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division 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, or PID is an Integral domain in which every ideal is principal i 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 set In Abstract algebra, a branch of Mathematics, an integral domain is a Commutative ring with an additive identity 0 and a multiplicative identity 1 such 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. In Mathematics, a field F is said to be algebraically closed if every Polynomial in one Variable of degree at least 1 with Coefficients Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted 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. In Mathematics, the real numbers may be described informally in several different ways For example, the spectrum of ℝ[t] consists of closed points (x − a), for a in ℝ, (x² + px + q) where p, q are in ℝ and with negative discriminant p² − 4q < 0, and finally a generic point (0). In Algebra, the discriminant of a Polynomial with real or complex Coefficients is a certain expression in the coefficients of the 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). Commutative algebra is the branch of Abstract algebra that studies Commutative rings their ideals, and modules over such rings In Commutative algebra, the Krull dimension of a ring R, named after Wolfgang Krull ( 1899 - 1971) is defined to be the In Commutative algebra, Krull's principal ideal theorem, named after Wolfgang Krull (1899 - 1971 gives a bound on the height of a Principal ideal

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. In Mathematics, in the fields of General topology and particularly of Algebraic geometry, a generic point P of a Topological space 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.

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. In Topology and related branches of Mathematics, a connected space is a Topological space which cannot be represented as the disjoint union of 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. In Algebraic geometry, a proper morphism between schemes is an analogue of a Proper map between Topological spaces Definition A In Mathematics, a scheme is an important concept connecting the fields of Algebraic geometry, Commutative algebra and Number theory.

Contents

The classical definition

Affine varieties

Projective varieties

Properties

The modern definition

Examples

Properties

See also