Truncated projective spaces, Brown-Gitler spectra and indecomposable A(1)-modules

TL;DR

This paper provides a comprehensive classification of indecomposable modules over the A(1) subalgebra of the Steenrod algebra, linking cohomology of Brown-Gitler spectra and truncated projective spaces to Ext-group calculations in Adams spectral sequences.

Contribution

It introduces a structure theorem for A(1)-modules, classifies a family of indecomposables, and unifies previous results on Ext-groups relevant to Adams spectral sequence computations.

Findings

Classification of indecomposable A(1)-modules

Identification of cohomology of Brown-Gitler spectra within the family

Unified approach to Ext-groups for Adams spectral sequence

Abstract

A structure theorem for bounded-below modules over the subalgebra A(1) of the mod 2 Steenrod algebra generated by Sq^1, Sq^2 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 spectra).