Commutative algebra final project 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!
 A: As someone who is not a true expert in commutative algebra but has written up a lot of notes, this question hits pretty close to home for me.  In fact writing the notes is in some ways performing the task you are asking of your students, several times over.
Perhaps you might be interested in some things which I wish were in my notes but currently are not, i.e., some projects of this type that I have not yet gotten around to.  Here are a few:

*

*A systematic discussion of Galois connections in commutative algebra (c.f. $\S 2$ of my notes)


*A discussion of the Stone-Cech compactification from the perspective of rings of continuous functions.  (c.f. $\S 5.2$ of my notes)


*A proof of Hochster's theorem, characterizing the topological spaces homeomorphic to the spectrum of some commutative ring as the inverse limits of finite Kolmogorov ("$T_0$") spaces. (c.f. $\S 13.7$ of my notes)


*A proof of the Shephard-Todd-Chevalley Theorem of classical invariant theory (c.f. $\S 
14.6$ of my notes)


*A proof of the Krull-Akizuki Theorem together with a discussion of the simplest possible examples of failure of finiteness of normalization. (c.f. $\S 18$ of my notes)


*A discussion of the interactions between model theory and commutative algebra -- e.g. a discussion of the fact that many non-Noetherian analogues of familiar properties turn out to be first order properties whereas the more familiar ones are not (e.g. Bezout domain versus PID).
A: 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. 
A: Idea 1: Sprinkled throughout Atiyah McDonald are over a dozen exercises concerning the structure of $Spec$.  Undoubtedly a great many of your students are taking commutative algebra because they are interested in algebraic geometry, and this would be particularly useful for them.  The challenge would be to come up with a "punchline" which ties the project together.
Idea 2: Those students who are not taking commutative algebra primarily to learn algebraic geometry are probably interested in number theory.  A nice project might be to prove from scratch that every prime in the ring of integers of a number field splits as the product of primes in the ring of integers of any finite algebraic extension.  If I recall correctly, this only uses basic facts about ideals and modules over Dedikind rings.  You might even be able to discuss a little bit of basic ramification theory.
A: You could ask them to discover (in effect) the spectral sequence for a specific composition of Exts and Tors, starting with a case where they can show that (say) Tor_n vanishes for n>1, and then proceeding to a case where Tor_n vanishes for n>2, (and so on for as far as you like) pointing them toward various interlocking exact sequences and prompting them to figure out how to organize all that information.
Then you could actually have them compute something specific using the general machinery they've developed.
An advantage of this project is that different students can be assigned different compositions of functors, giving problems that are different enough to discourage collaboration but similar enough to be graded on the same scale.
A: Some things I was thinking about myself:
*Comparing projective dimension, injective dimension, Krull dimension
*Proving some things on Gorenstein (local) rings
*Introducing height, depth and Cohen-Macaulay (local) rings
*Proving some special case of the Quillen-Suslin theorem (eg the two variables case, it's realistic)
*Introducing length and the Hilbert-Samuel polynomial and doing some concrete applications
*Discussing the basic properties of local cohomology
Any thoughts about this?
I think all of this is quite standard and I'd prefer to come up with something more original :)
