Quillen Stratification for the Steenrod Algebra

Abstract

Let A be the mod 2 Steenrod algebra. Its cohomology, H*(A;F2) = ExtA (F2, F2), is the E2-term of the Adams spectral sequence converging to the 2-component of ir~SS, the stable homotopy groups of spheres; as such, this cohomology algebra has been studied extensively for over thirty-five years. There have been a number of nice structural results and degree-by-degree computations, many of which have implications for -rxS0. Information about the ring structure of H* (A; F2), though, has been harder to come by. Analogously, in the study of the stable homotopy groups of spheres, there were many stem-by-stem calculations before Nishida proved in [14] that every element in E3i>o i7r SO was nilpotent. Since 7r8S0 Z and rS0 = O when i 0. In his landmark 1971 paper [18], Quillen gave a description of the cohomology algebra H* (G; k) modulo nilpotent elements as an inverse limit of the cohomology algebras of the p-subgroups of G. This has led, for instance, to the work of Benson, Carlson, et al. on the theory of varieties for kG-modules, and in general to deep structural information about modular representations of finite groups. In this paper we describe the cohomology of the Steenrod algebra, modulo nilpotent elements, according to the recipe in Quillen's result: as the inverse limit of the cohomology rings of the elementary abelian sub-Hopf algebras of A. We hope that this leads to further study in several directions. First of all, according to the point of view of axiomatic stable homotopy theory in [7], the representation theory of any cocommutative Hopf algebra has formal similarities to stable homotopy theory; viewed this way, our main result is an