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, Brown-Comenetz duality, and other homotopy-theoretic 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