 

Katharine Adamyk
Classifying and extending Q_{0} local A(1)modules
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Published: 
July 25, 2023. 
Keywords: 
Steenrod algebra, Margolis homology, localized Ext groups, DavisMahowald spectral sequence. 
Subject [2020]: 
55S10, 55T25, 16T05. 


Abstract
In the stable category of bounded below 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 [Bru14]. We show that all bounded below 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 A(1)modules. Each module in this family is comprised of copies of A(1)//A(0) linked by the action of Sq^{1} in A(1).
The classification theorem is then used to simplify computations of
h_{0}^{1}Ext_{A(1)}^{*, *}(, F_{2}) and to provide necessary conditions
for lifting A(1)modules to Amodules. We discuss a DavisMahowald spectral sequence converging to
h_{0}^{1}Ext_{A(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 Amodule. We give a formula for the second differential.


Acknowledgements
Portions of this work were supported by the National Science Foundation under grant No. DMS1906227.


Author information
Katharine Adamyk
Hamline University
Mathematics Department
1536 Hewitt Ave
St Paul, MN 55104, USA
kadamyk01@hamline.edu

