We apply the tools of stable homotopy theory to the study of mod￾ules over the Steenrod algebra A∗; in particular, we study the (triangulated)
category Stable(A) of unbounded cochain complexes of injective comodules
over A, the dual of A∗, in which the morphisms are cochain homotopy classes
of maps. This category satisfies the axioms of a stable homotopy category (as
given in [HPS97]); so we can use Brown representability, Bousfield localiza￾tion, Brown-Comenetz duality, and other homotopy-theoretic tools to study
Ext∗∗
A (Fp, Fp), which plays the role of the stable homotopy groups of spheres.
We also have nilpotence theorems, periodicity theorems, a convergent chro￾matic tower, and a number of other results