In the stable category of bounded below A(1)-modules, every module is determined by an extension between a module with trivial Q₀-Margolis homology and a module with trivial Q₁-Margolis homology [Bru14]. We show that all bounded below A(1)-modules of finite type whose Q₁-Margolis homology is trivial are stably equivalent to direct sums of suspensions of a distinguished family of A(1)-modules. Each module in this family is comprised of copies of A(1)//A(0) linked by the action of Sq¹ in A(1).

The classification theorem is then used to simplify computations of
h₀^-1Ext_A(1)^{*, *}(-, F₂) and to provide necessary conditions for lifting A(1)-modules to A-modules. We discuss a Davis-Mahowald spectral sequence converging to
h₀^-1Ext_A(1)^{*, *}(M, F_2) where M is any bounded below A(1)-module. The differentials in this spectral sequence detect obstructions to lifting the
A(1)-module, M, to an A-module. We give a formula for the second differential.

Portions of this work were supported by the National Science Foundation under grant No. DMS--1906227.