A Counterexample for Lightning Flash Modules over E(e1,e2)

TL;DR

This paper provides a counterexample to a specific theorem in Margolis's book, challenging assumptions about the structure of certain modules over the Steenrod algebra and impacting related proofs.

Contribution

It presents a counterexample to a key theorem, revealing limitations in the decomposition of modules over the Steenrod algebra.

Findings

Counterexample shows modules do not always split as lightning flash plus free modules

Implications for proofs relying on the original theorem

Highlights need for revised understanding of module structures

Abstract

We give a counterexample to Theorem 5 in Section 18.2 of Margolis' book, "Spectra and the Steenrod Algebra", and make remarks about the proofs of some later theorems in the book that depend on it. The counterexample is a module which does not split as a sum of lightning flash modules and free modules.