Truncated projective spaces, Brown–Gitler spectra and indecomposable [formula omitted]-modules

Abstract

A structure theorem for bounded-below modules over the subalgebra A(1) of the mod-2 Steenrod algebra generated by Sq1, Sq2 is proved; this is applied to prove a classification theorem for a family of indecomposable A(1)-modules. The action of the A(1)-Picard group on this family is described, as is the behaviour of duality.The cohomology of dual Brown–Gitler spectra is identified within this family and the relation with members of the A(1)-Picard group is made explicit. Similarly, the cohomology of truncated projective spaces is considered within this classification; this leads to a conceptual understanding of various results within the literature. In particular, a unified approach to Ext-groups relevant to Adams spectral sequence calculations is obtained, englobing earlier results of Davis (for truncated projective spaces) and recent work of Pearson (for Brown–Gitler spectrum).