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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!

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Community wiki? –  Igor Rivin Sep 14 '11 at 17:21
    
Yes, I agree that this should be community wiki. –  Wanderer Sep 14 '11 at 18:58

5 Answers 5

up vote 6 down vote accepted

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.

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Do you have a scan of these notes? Just curious to see them :) –  Wanderer Sep 15 '11 at 17:11
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I can try to scan the copy i have. i will be at my office next monday. unfortunately they are on non standard size (long) paper and might get shrunk. –  roy smith Sep 15 '11 at 21:03
    
Great, thanks a lot! –  Wanderer Sep 16 '11 at 1:29
    
ok i have the scan. i can email it. where to? there is a phrase missing at the top of page 15, maybe ("maximal ideal in the ring R_S") –  roy smith Sep 20 '11 at 4:52
    
I sent you an e-mail! –  Wanderer Sep 20 '11 at 17:20

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:

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

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

3) 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)

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

5) 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)

6) 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).

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I like the invariant theory idea! –  J.C. Ottem Sep 14 '11 at 20:30
    
I like suggestions 4 and 5 :) –  Wanderer Sep 14 '11 at 22:38

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.

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Thanks for the suggestion, I'll consider it! –  Wanderer Sep 14 '11 at 18:58
    
Or, less optimistically, just to study the effect of changing rings in a situation like going from $A$ to $A/(a)$ with $a$ a regular element. –  Mariano Suárez-Alvarez Sep 14 '11 at 20:52
    
The question and all other answers are CW, but this one is not because it was submitted before the OP edited the question to make it CW. This should also be edited to make it CW. –  David White Sep 14 '11 at 21:23
    
It's true that the project is quite ambitious... Can you give some more details of how you would organize this to make it accessible? –  Wanderer Sep 14 '11 at 22:37

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.

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@Paul: perhaps predictably, as an algebraic number theorist I worry that your second proposal is a little too "pat". There are plenty of texts which lead you by the hand from scratch to the factorization of ideals into primes in any Dedekind domain (and also prove that the integral closure of a Dedekind domain in a finite separable field extension is a Dedekind domain, so $\mathbb{Z}_K$ is one). If you don't have any background here, this could be 15-20 hours of work, but there's little or no synthesis required: e.g. copying out relevant passages from Samuel's little book would suffice... –  Pete L. Clark Sep 14 '11 at 20:01
    
Wouldn't having read Samuel's little book itself be a great thing to come out with from the project? –  Mariano Suárez-Alvarez Sep 14 '11 at 20:51
    
@Mariano: well, sure. It's just that (in particular) Samuel lays things out so nicely that this is more of a reading project than a writing project. –  Pete L. Clark Sep 14 '11 at 21:03
    
I agree that it's a bit dangerous, and moreover some of the students will be following a parallel class on algebraic number theory or algebraic geometry - even though they won't be doing schemes, so I'd have to think about it some more. –  Wanderer Sep 14 '11 at 22:36
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I am by no means a number theorist (or even an algebraist) and I don't think I've encountered the book of Samuel that you're referring to, so I'll defer to your judgement here. I brought up the example because it was particularly formative in my own experience with commutative algebra. I first started learning about algebraic number theory a bit after I had already taken a course on commutative algebra, and I vividly remember how a lot of the abstract machinery that had confused me the first time around finally "clicked" when I saw the proof that primes split in a finite algebraic extension. –  Paul Siegel Sep 15 '11 at 2:20

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 :)

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If you go the dimension route, why not discuss right-global dimension and weak-dimension of a ring? A great reference is T.Y. Lam's Lectures on Modules and Rings. There are plenty of examples and some (hard) open problems. –  David White Sep 15 '11 at 0:03
    
That project gives you a nice way to get into Ext and Tor also. By the way, a problem I've been trying to solve on and off (which so far has used a lot from Lam, but without success) is posted here: mathoverflow.net/questions/61274/… –  David White Sep 15 '11 at 0:05

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