# 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?

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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?)

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 except perhaps for some basic commutative algebra results like the Nullstellensatz, my knowledge of algebraic geometry is practically null. – Holdsworth88 Aug 30 '12 at 19:05

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.