  We present progress in trying to verify a long-standing conjecture by Mark
Mahowald on the $v_{1}$-periodic component of the classical Adams spectral
sequence for a Moore space $M$. The approach we follow was proposed by John
Palmieri in his work on the stable category of $A$-comodules. We improve on
Palmieri's work by working with the endomorphism ring of $M$ - $End(M)$ thus
resolving some of the initial difficulties of his approach and formulating a
conjecture of our own that would lead to Mahowald's formulation.
