Zariski topology

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

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

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

• 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 previous one, defines the Zariski topology on any affine variety.

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 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 "homogeneous ideal", 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.

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

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 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 (x² + px + q) where p, q are in ℝ and with 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).

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 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.