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There is a standard syllabus for a first graduate course in algebra. One teaches groups, rings, fields, perhaps a little bit of Galois theory, perhaps a little bit of category theory, perhaps a little bit of representation theory, all this a little bit superficially, to give an idea of the fundamental algebraic structure to graduate students that will work in all parts of mathematics.

I have much more difficulties to see what to teach in a second, more advanced, course in algebra, whose student body is constituted of the grad students who like algebra, whatever they are eventually going to work in. Commutative algebra is excluded because in my department, as in many others, there is another course devoted to this specific subject. But even so, there are so many loosely inter-related things (more category theory, more homological algebra, more representation theory, advanced theory of finite groups, study of classical groups, theory of groups defined by generators and relations, Brauer theory, etc.) one could think of that I find very difficult to arbitrage between them. One is naturally pushed to give a course with no unity, which is not very pleasant.

Since the problem I experience has certainly been met by others, I'd like to know: What did you or would you teach in such a course? What are the subjects that are absolutely necessary to teach (if any)? How to give the course a backbone? What textbook to use?

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I think homological algebra, category theory and some representation theory is suitable for this course. –  user16974 Oct 7 '11 at 17:11
Here, there is no second course in algebra, just topics (presumably for the reasons you stated). However there is a rep theory course, a homological algebra course, etc., etc. If there isn't a set curriculum in place, it must not be an established course; would it be possible to just teach an approachable topic course instead? To go deeper into any one subject than is taught in a first course seems to justify an entire course on its own. –  Richard Rast Oct 7 '11 at 17:25
Are your first course and second course each one semester or two semesters? Is such a second course about to be introduced in your department (I presume it isn't already there or you would have told us what past instructors in it did) or is this more of a hypothetical question? As for topics, if your list from a first course is accurate then I'd say a second course should at least discuss linear and multilinear algebra. –  KConrad Oct 7 '11 at 18:09
In departments with modest-sized graduate programs it's rare to have a "second course" in any graduate-level subject. My department for example mainly tries to offer a variety of one-semester "topics" courses depending on the range of faculty research interests. There are usually too many courses proposed, but however it's done the student should be weaned from over-reliance on introductory textbooks even if one or more books are recommended as back-ups. The point is not just to go on taking courses forever in a passive mode but rather to get actively involved in current problems. –  Jim Humphreys Oct 7 '11 at 18:58
Perhaps some general algebra viewpoints along with a standard treatment of some of the topics suggested by others? While covering standard literature, results, and methods are useful in introducing students to a particular area which has been well researched and developed decades ago, would it not be as good to show different viewpoints of the material that they can apply or tweak to help them make new mathematics? Gerhard "Showing Some General Algebra Bias" Paseman, 2011.10.08 –  Gerhard Paseman Oct 8 '11 at 7:22
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3 Answers

As a graduate student in algebraic topology, but one who has taken many "second year" graduate courses in algebra, the one I think I would have enjoyed the most had it ever been offered (and the one which would have been most useful for me personally) would go something like this:

Textbook: An Introduction to Homological Algebra - Charles A. Weibel

What to cover:

  • Chain complexes and homology
  • Derived functors, Ext, and Tor
  • Spectral Sequences and/or homological dimension depending on which direction you want to go
  • Group Homology and Cohomology (I really enjoy Weibel's treatment of this)
  • Lie Algebra Homology and Cohomology (here you can bring in lots of related topics)
  • Last chapter and appendices on category theory and the derived category

I agree with Richard Rast a bit that no one course can cover all the topics you like, but I think Weibel does a great job setting up the homology/cohomology framework using category theory and lots of homological algebra, applying this machinery to group cohomology and representation theory, and also bringing in classical groups. This seems to cover most of what you mention in your question.

A supplement I used when following this model on my own was Representations and Cohomology Parts I and II by D.J. Benson

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This seems to be a topics course or "homological algebra" course, which I would say is distinct from what one would normally consider a 2nd semester of Algebra. –  Chris Gerig Oct 7 '11 at 19:18
@Chris Gerig: I know it misses a lot of topics one might like, but Weibel's book also seems to be a closer approximation to what the OP wanted than anything else I could think of. The bits about group cohomology, Lie groups, and representation theory for example aren't what I would call pure homological algebra. It would be nice to get a mention of quadratic forms, elliptic curves, modular forms, or whatever else in there, but I really don't think one course can coherently cover all this material. The course I outline covers a lot and is coherent, but I don't claim it's perfect. –  David White Oct 7 '11 at 20:22
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I'm posting a separate answer because I realized my first might be too much algebraic topology. Another great second year course idea is to follow Lectures on Modules and Rings by T.Y. Lam. This book is my bible for homological algebra, and I have never heard anyone claim there was a better book for this. The only "downside" is that everything is non-commutative, but Lam does a great job of telling you exactly what commutativity gets you (via corollaries to the theorems), so even students who go on to work primarily in a commutative setting will not be ill-served.

If you wanted to cover lots of great homological algebra in this proposed second course (without an eye towards algebraic topology), I can't think of a better book. Here are some topics:

  • Projective, Injective, and Flat Modules
  • Semisimple, Coherent, Von Neumann Regular Rings, Cohen-Macaulay, and Gorenstein Rings
  • Homological Dimensions and Regular Local Rings
  • Localization
  • Quasi-Frobenius Rings and Algebras
  • Matrix Rings and representation theory

Since this is the "second course" to his "First Course in Noncommutative Rings" one might be tempted to use that text for the first course. I'm not sure this is such a good idea. While I love Lam's writing style and the vast amount of material he covers, it seems a lot of what A First Course covers isn't really necessary to do algebra later on, i.e. a lot of it deals with situations which modern research avoids via standard assumptions on the rings in question.

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I don't own this book and haven't really read it -- I've only consulted it once or twice for some specific purpose. But are you sure you'd claim there is no better book for homological algebra? Sure, there's lots of overlap with homological concepts, but looking at the table of contents, there's nothing on derived functors, nothing on derived categories, nothing on spectral sequences -- it doesn't qualify as a book on homological algebra per se, as far as I can tell. –  Todd Trimble Oct 7 '11 at 21:04
@Todd Trimble: I certainly should not claim there's no better book for homological algebra in general. You're right that Lam is lacking in the categorical side. I guess I meant: "For what it covers, there is no better book" (and I shouldn't even claim this, since I've only read a few books that try to cover this material). I was just trying to compliment Lam for how complete and well-written this book is. –  David White Oct 7 '11 at 22:29
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When I took the second graduate algebra course we spent the first half on noncommutative ring theory. We covered topics like Jacobson radicals, artinian/noetherian rings, semi-simple algebras. We used Herstein and Jacobson's algebra books. The second half was all the ground work for complex representations of finite groups. No text was suggested, but it was a bit like "Linear Representations of Finite Groups" by Serre.

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