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.

(more…)

Read Full Post »