Module Theory for the Working Student
Question:
What level of familiarity and comfort with modules should someone looking to work through Hatcher's Algebraic Topology possess?
Motivation:
I am taking my first graduate course in Algebraic Topology this coming October. The general outline of the course is as follows:
- Homotopy, homotopy invariance, mapping cones, mapping cylinders
- Fibrations, cofibrations, homotopy groups, long-exact sequences
- Classifying spaces of groups
- Freudenthal, Hurewicz, and Whitehead theorems
- Eilenberg-MacLane spaces and Postnikov towers.
I have spent a good deal of this summer trying to fortify and expand the foundations of my mathematical knowledge. In particular, I have been reviewing basic point-set and algebraic topology and a bit of abstract algebra. My knowledge of module theory is a bit lacking, though. I've only covered the basics of the following topics: submodules, algebras, torsion modules, quotient modules, module homomorphisms, finitely generated modules, direct sums, free modules, and a little bit about $\text{Hom}$ and exact sequences, so I have a working familirity with these ideas. As I do not have much time left before the beginning of the semester, I am trying to make my studying as economical as possible. So perhaps a more targeted question is:
What results and topics in module theory should every student in Algebraic Topology know?
Honestly, I don't think you need to know much more beyond the basics. Hatcher treats everything from as elementary a standpoint as possible. In particular, he covers the algebra as it's needed. I certainly would not say you need to know algebraic geometry. It sounds like you already know all you need, although it couldn't hurt to know the statement of the classification of modules over PIDs.
I read Algebraic Topology by Hatcher before I really knew what it meant for a module to be flat, or projective . I of course knew what a module was, but besides that, I don't think there's much more needed.
The arguments (as I remember them) in the books where you need to know about modules comes from homology or cohomology. It is crucial that you know what the kernel of a map is, and can spot short exact sequences (and know why they're useful).
Maybe there's some deeper theorems in the chapter on cohomology, but I think you could probably try to learn that as you go along.
So, in short, I think you should just go ahead and read it. If you encounter problems, just ask here and we'll be sure to help you!
Looks like you'll definitely want to know the homological aspects: projective, injective and flat modules. There are lots of different characterizations for these which are useful to know. (Sorry, don't know the insides of D&F very well, perhaps it is covered.)
The Fundamental theorem of finitely generated modules over a principal ideal domain is a pretty good one to know too, since it kind of bundles up the entire theory of linear algebra.
(How much algebraic geometry do you know?)