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 co modules 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.
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Zielgruppe
Für höhere Schule und Studium
Für Beruf und Forschung
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ISBN-13
978-0-8218-2668-3 (9780821826683)
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Schweitzer Klassifikation
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.
