Zariski topology

In

of the ring) a topological space.

The Zariski topology allows tools from

.

The Zariski topology of an algebraic variety is the topology whose

, 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

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

).

Affine varieties

First, we define the topology on the affine space ${\displaystyle \mathbb {A} ^{n},}$ formed by the

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 ${\displaystyle \mathbb {A} ^{n}.}$ 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. ${\displaystyle V(I)\cup V(J)\,=\,V(IJ);}$
  2. ${\displaystyle V(I)\cap V(J)\,=\,V(I+J).}$

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). This is the Zariski topology on ${\displaystyle \mathbb {A} ^{n}.}$

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 ${\displaystyle \mathbb {A} ^{n}.}$ Equivalently, it can be checked that:

• The elements of the affine
  coordinate ring
  $${\displaystyle A(X)\,=\,k[x_{1},\dots ,x_{n}]/I(X)}$$

  
  act as functions on X just as the elements of ${\displaystyle k[x_{1},\dots ,x_{n}]}$ act as functions on ${\displaystyle \mathbb {A} ^{n}}$; 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
  $${\displaystyle V'(T)=\{x\in X\mid f(x)=0,\forall f\in T\}}$$

  
  (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 definition of the closed sets in ${\displaystyle \mathbb {A} ^{n}}$ above, defines the Zariski topology on any affine variety.

Projective varieties

Recall that n-dimensional projective space ${\displaystyle \mathbb {P} ^{n}}$ is defined to be the set of equivalence classes of non-zero points in ${\displaystyle \mathbb {A} ^{n+1}}$ by identifying two points that differ by a scalar multiple in k. The elements of the polynomial ring ${\displaystyle k[x_{0},\dots ,x_{n}]}$ are not generally functions on ${\displaystyle \mathbb {P} ^{n}}$ 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

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

The same facts as above may be established for these sets, except that the word "ideal" must be replaced by the phrase "

", so that the V(S), for sets S of homogeneous polynomials, define a topology on ${\displaystyle \mathbb {P} ^{n}.}$ 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

.

By Hilbert's basis theorem and the fact that Noetherian rings are closed under quotients, 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 (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.

Every

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 ${\displaystyle \mathbb {A} ^{1}.}$

Spectrum of a ring

In modern algebraic geometry, an algebraic variety is often represented by its associated

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

${\displaystyle V(I)=\{P\in \operatorname {Spec} A\mid P\supseteq I\}}$

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 (a₁, ..., a_n) such that the ideal generated by the polynomials 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. 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:

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

("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 we have

${\displaystyle e_{a}(P)=0\Leftrightarrow P\in V(a)}$

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. 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 ${\displaystyle \mathbb {Z} }$, the spectrum of the integers has a closed point for every prime number p corresponding to the maximal ideal ${\displaystyle (p)\subseteq \mathbb {Z} }$, 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 ${\displaystyle \mathbb {Z} }$ are precisely the whole space and the finite unions of closed points.
• Spec k[t], the spectrum of 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 ${\displaystyle \mathbb {R} [t]}$ consists of the closed points (x − a), for a in ${\displaystyle \mathbb {R} }$, the closed points (x² + px + q) where p, q are in ${\displaystyle \mathbb {R} }$ and with negative
  prime factorization
  of a generator of the ideal).

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

T₀ spaces

: given two points P, Q that 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 topological 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.

See also

• Spectral space

Citations

References