I'm looking for a topic for a final project in commutative/homological algebra, for first year master's students (in a decent European university). During the course, they will cover the following topics: commutative rings (as in chapter 1 of Atiyah-McDonald), general module theory and structure of finitely generated modules over a PID, tensor products, basic category theory - including, products, coproducts, Yoneda - complexes and (co)homology, derived functors, flat, injective and projective modules, first properties of Tor and Ext. Not included will be: Artinian modules and length, notions of dimension, completion, localization, valuation rings, regular local rings (though DVR's, Nakayama are fine).
The goal would be to prove some nice result and possibly introduce some new notions along the way - everything in the form of (rather long) series of exercises - without having to develop too much big machinery or new theory. The project should take about 15-20 hours of work. Of course the topic could be (part of) one of the topics which I mentioned above as "not treated in class". Ideally it should be a "synthesis" and use a lot of the techniques learned in the course.
Any suggestions? Classical theorems, things extracted from recent research...
I'm looking forward to your suggestions!