The object of study for this book is the mod p Steenrod algebra A and its cohomology Ext(A)**(F-p, F-p). Various people, including the author, have approached this subject by taking results in stable homotopy theory and then trying to prove analogous results for A-modules. This has proven to be successful, but the analogies were just that-there was no formal setting in which to do anything more precise. In [HPS97], Hovey, Strickland, and the author developed "axiomatic stable homotopy theory." In particular, we gave axioms for a stable homotopy category; in any such category, one has available many of the tools of classical and modern stable homotopy theory-tools like Brown representability and Bousfield localization. It turns out that a category Stable(A) (defined in the next paragraph) of modules over the Steenrod algebra is such a category; as one might expect, the trivial module F, plays the role of the sphere spectrum S-0, and Ext(A)**(-, -) plays the role of homotopy classes of maps. Since many of the tools of stable homotopy theory are focused on the study of the homotopy groups of S-0 land of other spectral, one should expect the corresponding tools in Stable(A) to help in the study of Ext(A)**(F-p, F-p) land related groups). In this book we apply some of these tools-Adams spectral sequences, nilpotence theorems, periodicity theorems, chromatic towers, etc.-to the study of Ext over the Steenrod algebra. It is our hope that this book will serve two purposes: first, to provide a reference source for a number of results about the Steenrod algebra and its cohomology, and second, to provide an example of an in-depth use of the language and tools of axiomatic stable homotopy theory in an algebraic setting. Now we describe the category in which we work. We fix a prime p, let A* be the mod p Steenrod algebra, and let A = Hom(Fp)(A*,Fp) be the Igraded) dual of the Steenrod algebra. We let Stable(A) be the category whose objects are unbounded cochain complexes of injective left A-comodules, and whose morphisms are cochain homotopy classes of maps. This is a stable homotopy: category We prove a number of results about Stable(A); some of these are analogues of results in the ordinary stable homotopy category, and some are not. Some of these are new, and some already known, at least in the setting of A*-modules; the old results often need new proofs to apply in the more general setting we discuss here.
