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

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 spectrum).2000 Mathematics Subject Classification. 19L41; 55S10. Key words and phrases. Steenrod algebra, indecomposable module, truncated projective space, Brown-Gitler spectra.The author is very grateful to an anonymous referee for remarks which have lead to an improvement in the presentation of the paper.

The stable Picard group of Hopf algebras via descent, and an application

“…Using our technique, it is also possible to recover and extend the classification of lifts of E(1)-modules to A(1)-modules from [Pow15]. However, in loc cit, the author needed an indecomposability hypothesis which is not needed for our approach.…”

“…This last question was already studied by Geoffrey Powell in [Pow15]. In loc cit, he is able to compute by hand the number of lifts of each indecomposable E(1)-module.…”