The Steenrod algebra is self-injective, and the Steenrod algebra is not self-injective

Abstract

It is well-known that the Steenrod algebra $A$ is self-injective as a graded ring. We make the observation that simply changing the grading on $A$ can make it cease to be self-injective. We see also that $A$ is not self-injective as an ungraded ring. These observations follow from the failure of certain coproducts of injective $A$-modules to be injective. Hence it is natural to ask: which coproducts of graded-injective modules, over a general graded ring, remain graded-injective? We give a complete solution to that question by proving a graded generalization of Carl Faith's characterization of $\Sigma$-injective modules. Specializing again to the Steenrod algebra, we use our graded generalization of Faith's theorem to prove that the covariant embedding of graded $A_*$-comodules into graded $A$-modules preserves injectivity of bounded-above objects, but does not preserve injectivity in general.