We’re going to change topics again, but we’re sticking to the theme of doing things that just barely don’t get covered in a first-year graduate algebra course (maybe some of these things do get covered at some universities) but turn out to be extremely useful in “real life.” Today we’ll start a series of posts on modules over Dedekind domains.
The reason this turns out to be useful is that many examples in algebraic/arithmetic geometry require you to look no further than understanding modules over Dedekind domains. The other extremely useful place this comes up is in algebraic number theory where the integral closure of in a number field , called the ring of integers of , is a Dedekind domain. It turns out that understanding these is an important part of classical algebraic number theory. This is important for more modern trends with modular forms and elliptic curves as well, since these appear as the endomorphism ring of elliptic curves with complex multiplication.
I recently spent a long time trying to work something out that just turned out to be standard known theory of modules over Dedekind domains and this motivated me to do this series next. It seems to me that graduate courses do the structure theorem for finitely generated modules over PIDs, and talk about Dedekind domains, but that is where they leave off which means that is where we’ll start.
Recall a Dedekind domain is an integrally closed, Noetherian domain of dimension . There are lots of other characterizations as well, but we’ll just recall those if they are needed. Our main goal is to prove a structure theorem for finitely generated projective modules. We’ll start with something more basic that should be familiar from the PID case. Fix a Dedekind domain and a finitely generated -module. It turns out that is projective if and only if it is flat if and only if it is torsion free.
Maybe this is cheating to assume “first year algebra” to prove this, since my cut-off is quite arbitrary, but torsion free if and only if flat follows immediately by noticing these are local statements. Since the localization of a Dedekind domain is a DVR which is a PID, the equivalence follows from the fact that a finitely generated module is torsion free if and only if it is free over a PID.
Flat always implies projective (under reasonable assumptions). Thus the only thing left to check is that torsion free implies projective. Suppose is torsion free. Note that it is projective if and only if all sequences split. If is projective we can just use the universal property with the identity map to obtain a section. If there is section, just take a finite presentation by free modules to realize as a direct summand of a free module.
Now suppose we’re given such a sequence. After any localization it certainly splits because again we have these results for PIDs already. The idea of the rest of the proof is to glue these local splittings into an honest section . Suppose our local maps are . Since we are finitely generated we can look at the denominators of the image of the generators under and multiply them together to give an element such that .
Now we make our map by choosing finitely many and elements such that . Our section is obtained by “gluing” to make , by . Now it is just checking that this works, so we won’t do it. The setup was primed to make it work. Thus our three conditions are equivalent.
Now our module can be broken up into a torsion part and a projective part, i.e. . As with the PID case the torsion part with the appropriate uniqueness statement. Thus we do get a nice similar structure theorem, but as I said the part that is really interesting for the future application I want to point out is the structure of the projective part.