New York Journal of Mathematics
Volume 29 (2023), 939-980


Katharine Adamyk

Classifying and extending Q0 local A(1)-modules

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Published: July 25, 2023.
Keywords: Steenrod algebra, Margolis homology, localized Ext groups, Davis--Mahowald spectral sequence.
Subject [2020]: 55S10, 55T25, 16T05.

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
h0-1ExtA(1)*, *(-, F2) and to provide necessary conditions for lifting A(1)-modules to A-modules. We discuss a Davis-Mahowald spectral sequence converging to
h0-1ExtA(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.

Author information

Katharine Adamyk
Hamline University
Mathematics Department
1536 Hewitt Ave
St Paul, MN 55104, USA