If you haven’t heard the terms in the title of this post, then you are probably bracing yourself for this to be some weird post on innuendos or something. Let’s first do some motivation (something I’m not often good at…remember that Jacobson radical series of posts? What is that even used for? Maybe at a later date we’ll return to such questions). We can do ring extensions just as we do field extensions, but they tend to be messier for obvious reasons. So we want some sort of property that will force an extension to be with respect to prime ideals. Two such properties are “lying over” and “going up.”
Let be a ring extension. Then we say it satisfies “lying over” if for every prime ideal in the base, there is a prime ideal in the extension such that . We say that satisfies “going up” if in the base ring are prime ideals, and lies over , then there is a prime ideal which lies over . (Remember that Spec is a contravariant functor).
Note that if we are lucky a whole bunch of posts of mine will finally be tied together and this was completely unplanned (spec, primality, localization, even *gasp* the Jacobson radical). First, let’s lay down a Lemma we will need:
Let be an integral extension of R. Then
i) If a prime ideal of R and lies over , then is integral over .
ii) If , then is integral over .
Proof: By the second iso theorem , so we can consider as a subring of . Take any element . By integrality there is an equation with the . Now just take everything to get that integral over . This yields part (i).
For part (ii), let , then , where and . By integrality again we have that , so we multiply through by in the ring of quotients to get . Thus is integral over .
I’ll do two quick results from here that will hopefully put us in a place to tackle the two big results of Cohen and Seidenberg next time.
First: If is an integral ring extension, then is a field if and only if is a field. If you want to prove this, there are no new techniques from what was done above, but you won’t explicitly use the above result, so I won’t go through it.
Second: If is an integral ring extension, ten if is a prime ideal in R and is a prime ideal lying over , then is maximal if and only if is maximal.
Proof: By part (i) of above, is integral over and so as a corollary to “First” we have one is a field if and only if the other is. This is precisely the statement that is maximal iff is maximal.