Stable Homotopy over the Steenrod Algebra, PDF eBook

Stable Homotopy over the Steenrod Algebra PDF

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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.

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