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Stable Homotopy over the Steenrod Algebra
John H. Palmieri, University of Washington, Seattle, WA
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Memoirs of the American Mathematical Society
2001; 172 pp; softcover
Volume: 151
ISBN-10: 0-8218-2668-9
ISBN-13: 978-0-8218-2668-3
List Price: US$62 Individual Members: US$37.20
Institutional Members: US\$49.60
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, 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.

Graduate students and research mathematicians interested in algebraic topology.

• Appendix B. Steenrod operations and nilpotence in $$\mathrm{Ext}_\Gamma^{**}(k,k)$$