  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.
