For commutative unital C*-algebras and for finitely-generated reduced integral -algebras, we have seen that is a functor which sends homomorphisms to continuous functions. However, this is not true for general commutative rings. What we want is for a ring homomorphism to be sent to a continuous function
via contraction. Unfortunately, the contraction of a maximal ideal is not always a maximal ideal. The issue here is that a maximal ideal of is just a surjective homomorphism where is some field, and the contracted ideal is just the kernel of the homomorphism . However, this homomorphism need no longer be surjective, so it may land in a subring of which may not be a field. For a specific example, consider the inclusion . The ideal is maximal in , but its contraction is the ideal in , which is prime but not maximal.
In other words, if we want to think of ring homomorphisms as continuous functions on spectra, then we cannot work with maximal ideals alone. Prime ideals are more promising: a prime ideal is just a surjective homomorphism where is some integral domain, and the contracted ideal of a prime ideal is always prime because a subring of an integral domain is still an integral domain. Now, therefore, is an appropriate time to replace with , the space of all prime ideals equipped with the Zariski topology, and this time is a legitimate contravariant functor .
In this post we’ll discuss this choice. I should mention that the Secret Blogging Seminar has discussed this point very thoroughly already, but from a much more high-brow perspective.
It’s possible this does not need to be said, but for the record, the Zariski topology on is defined in exactly the same way as on : there is a Galois connection sending a subset of to the ideal of all elements of vanishing on them and sending a subset of to the set of all prime ideals on which they vanish, and the composition defines the closure operator on . Equivalently, the closed sets of are the vanishing sets of subsets of . Essentially all of the basic properties of the Galois connection we proved for maximal ideals carries over, and in a sense the ideal-variety correspondence is nicer in this case because now always holds without the Jacobson assumption.
I’m not completely certain how I’m going to distinguish between the two versions of . Hopefully it’ll be clear from context.
Why should we care?
For finitely-generated reduced integral -algebras, prime ideals are naturally in bijection with subvarieties, so it is possible to recover any information about prime ideals from maximal ideals alone. More generally, for any Jacobson ring the prime ideals can be identified with the irreducible closed subsets of . So at least in this restricted case it is not at all necessary to consider the full spectrum.
However, prime ideals are important when we want to work in more general rings, and even to study varieties in we’ll often want to use rings which are not finitely generated, such as power series rings. The maximal spectrum of a power series ring is a point, a manifestation of the fact that when we work in a power series ring we’re restricting our attention to a single point, but this is clearly an inadequate description of a power series ring; it doesn’t even tell us how many variables we’re dealing with. Prime ideals go some way towards rectifying the situation.
For example, has unique maximal ideal and exactly one additional prime ideal . has unique maximal ideal , but it also has many additional prime ideals such as those of the form where are units. Since should be thought of as the ring of germs at the origin, these prime ideals can be thought of as representing various ways in which a curve in can approach the origin.
Another important reason to work with the prime spectrum, but one that won’t be too relevant for us, is the need in arithmetic geometry to switch between non-algebraically closed fields and even non-fields such as . As we already saw, switching between and requires thinking about prime ideals.
While it is reasonable to picture maximal ideals as distinct points because they can never contain each other, prime ideals are more complicated because they can. For example, the prime ideals of for an affine variety include the points of , but they also include extra non-maximal prime ideals, one for every subvariety of , including the point corresponding to itself. In topological terms, the non-maximal points of aren’t closed, since their closure contains all of their subvarieties. We say that they are “generic points” of the subvarieties they determine, and in diagrams generic points are visualized as points “smeared out” over these varieties. (This terminology is inherited from the history of the Italian school of algebraic geometry.)
One way to think about this situation is that is a kind of incidence geometry generalizing the geometry of points, lines, planes, and so forth in affine space. Points of have a partial ordering by closure (or equivalently, containment); we’ll say that for points if the ideal strictly contains the ideal . If is Noetherian, an ascending chain of ideals must terminate, which leads to the following important definition.
Definition: The Krull dimension of a Noetherian ring is the supremum of the number of strict inclusions in any chain of prime ideals of , if it exists.
Equivalently, the Krull dimension of a Noetherian space is the supremum of the number of strict inclusions in any chain of irreducible closed subsets, again if it exists, and the Krull dimension of is the Krull dimension of . The Krull dimension need not exist, even for Noetherian spaces but it will in all the cases we care about.
One way to motivate this definition is that the dimension of a vector space is the supremum of the number of strict inclusions in any chain of subspaces, since in the maximal case each strict inclusion corresponds to a reduction of the dimension by one. In fact, given a vector space of dimension the symmetric algebra is isomorphic to the polynomial algebra in indeterminates, and it turns out that the Krull dimension of this ring is as expected, so the Krull dimension is a strict generalization of the dimension of a vector space. This agrees with our intuition that should have dimension . More generally, the Krull dimension of is the dimension of the variety .
Fields have Krull dimension zero, which implies that has Krull dimension zero whenever is a maximal ideal. This agrees with our intuition that maximal ideals correspond to points, which are zero-dimensional phenomena, in .
We have seen before that prime factorization looks strange in rings where nonzero prime ideals are allowed to contain each other. The problem is one of Krull dimension: for example, in , which has Krull dimension two, it is unclear whether a prime ideal such as should have a “prime factorization” consisting of itself or consisting of the maximal ideals containing it. To avoid these issues, when we discuss factorization again we will restrict our attention to the case of Krull dimension one, which includes , and (it will turn out) their finite extensions. An affine variety of Krull dimension one is exactly an algebraic curve, and so we begin to see that there might be something to be gained from comparing rings of integers to algebraic curves.