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!
Best Answer
this is similar to some other answers.
when i took basic graduate algebra from Maurice Auslander he handed out 16 pages of very terse notes the first day that he said was our Fall semester final exam. There were four sections and each of us was assigned to read, learn and write up in more detail one section. They were on i) depth, ii) modules of finite projective dimension, iii) regular local rings, iv) unique factorization domains.
To give an idea of the style, the first sentence defined M depth N (modules over any ring) to be (when finite) the smallest degree such that Ext(M,N) is non zero. One page later he proved this integer (if finite) equals the length of a maximal N regular rad(A) sequence where A = ann(M), if R is noetherian and M,N finitely generated.
In the second section he defined projective dimension and related it for fin gen modules over noetherian local rings to the length of a minimal free resolution and the non vanishing of Tor. He then proved the formula relating depth and the dimensions of R,M. He used without proof facts such as tensoring with flat algebras commutes with Ext and (if faithfully flat) leaves projective dimension unchanged.
In section 3 he characterized when noetherian local rings are regular in terms of global dimension, projective dimension of modules, and regular sequences, and equated global dimension with krull dimension for such rings.
In the last section he showed every regular local ring is ufd, and the formal power series ring over any regular ufd is also regular ufd.
I did not yet know what an ideal was when I started the semester. You never forget a class like that.