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?
Best Answer
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:
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