The mod-2 Steenrod algebra A arises topologically as the algebra of stable operations on cohomology with coefficients in Z/2. There is a family of sub- Hopf algebras, A(n), that filter A. Given a module over A(n), one natural question to ask is whether it is the underlying A(n)-module for some A-module. (If it is, we say the A(n)-module lifts to an A-module.) In this talk, we approach the lifting question for the case n=1 via a classification theorem for a certain class of stable A(1)-modules.

In general, the stable category of modules over some ring, R, is obtained by factoring out the projective modules. When R is A(1), this is an especially nice category, and we will spend some time talking about its properties. We will also discuss Margolis homology, a useful invariant for A(1)-modules and its utility in the context of stable modules. The classification theorem describes A(1)-modules satisfying a condition on the Margolis homology groups.

Finally, we will discuss an application of the classification theorem to computing localized Adams spectral sequences and to the lifting problem.