On $P$-algebras and their duals

TL;DR

This paper advances the theory of $P$-algebras, focusing on their modules, duals, and cohomology calculations, with applications to topology.

Contribution

It develops systematic techniques for computing cohomology of $P$-algebras and their duals, including vanishing results and applications.

Findings

Techniques for calculating cohomology groups over $P$-algebras

Vanishing results for cohomology in specific cases

Examples illustrating topological applications

Abstract

The notion of $P$-algebra due to Margolis, building on work of Moore and Peterson, was motivated by the case of the Steenrod algebra at a prime and its modules. We develop aspects of this theory further, focusing especially on coherent modules and finite dimensional modules. We also discuss the dual Hopf algebra of $P$-algebra and its comodules. One of our aims is provide a collection of techniques for calculating cohomology groups over $P$-algebras and their duals, in particular giving vanishing results. Much of our work is implicit in that of Margolis and others but we are unaware of systematic discussions in the literature. We give some examples illustrating topological applications which follow easily from our results.