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?