Memoirs of the American Mathematical Society 2001; 172 pp; softcover Volume: 151 ISBN10: 0821826689 ISBN13: 9780821826683 List Price: US$66 Individual Members: US$39.60 Institutional Members: US$52.80 Order Code: MEMO/151/716
 We apply the tools of stable homotopy theory to the study of modules over the mod \(p\) Steenrod algebra \(A^{*}\). More precisely, let \(A\) be the dual of \(A^{*}\); then we study the category \(\mathsf{stable}(A)\) of unbounded cochain complexes of injective comodules over \(A\), in which the morphisms are cochain homotopy classes of maps. This category is triangulated. Indeed, it is a stable homotopy category, so we can use Brown representability, Bousfield localization, BrownComenetz duality, and other homotopytheoretic tools to study it. One focus of attention is the analogue of the stable homotopy groups of spheres, which in this setting is the cohomology of \(A\), \(\mathrm{Ext}_A^{**}(\mathbf{F}_p,\mathbf{F}_p)\). We also have nilpotence theorems, periodicity theorems, a convergent chromatic tower, and a number of other results. Readership Graduate students and research mathematicians interested in algebraic topology. Table of Contents  Preliminaries
 Stable homotopy over a Hopf algebra
 Basic properties of the Steenrod algebra
 Chromatic structure
 Computing Ext with elements inverted
 Quillen stratification and nilpotence
 Periodicity and other applications of the nilpotence theorems
 Appendix A. An underlying model category
 Appendix B. Steenrod operations and nilpotence in \(\mathrm{Ext}_\Gamma^{**}(k,k)\)
 Bibliography
 Index
