What to teach in a second graduate course in algebra? What textbook to use? 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?
 A: 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
A: 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.
A: 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.
