Classifying and extending $Q_0$-local $\mathcal{A}(1)$-modules

Abstract

In the stable category of bounded below $\mathcal{A}(1)$--modules, every module is determined by an extension between a module with trivial $Q_0$-Margolis homology and a module with trivial $Q_1$-Margolis homology. We show that all bounded below $\mathcal{A}(1)$-modules of finite type whose $Q_1$-Margolis homology is trivial are stably equivalent to direct sums of suspensions of a distinguished family of $\mathcal{A}(1)$-modules. Each module in this family is comprised of copies of $\mathcal{A}(1)//\mathcal{A}(0)$ linked by the action of $S{q}^1\in \mathcal{A}(1)$. The classification theorem is then used to simplify computations of $h_0^{-1}{\mathrm{Ext}}_{\mathcal{A}(1)}^{\bullet ,\bullet }(-,{\mathbb{F}}_2)$ and to provide necessary conditions for lifting $\mathcal{A}(1)$-modules to $\mathcal{A}$-modules. We discuss a Davis--Mahowald spectral sequence converging to $h_0^{-1}{\mathrm{Ext}}_{\mathcal{A}(1)}^{\bullet ,\bullet }(M,{\mathbb{F}}_2)$ where $M$ is any bounded below $\mathcal{A}(1)$-module. The differentials in this spectral sequence detect obstructions to lifting the $\mathcal{A}(1)$-module, $M$, to an $\mathcal{A}$-module. We give a formula for the second differential.

37 pages, 8 figures