Unstable modules with only the top k Steenrod operations

Abstract

We study an abelian category of unstable modules with the top $k$ Steenrod operations at the prime 2. We show that this category has homological dimension at most $k$. We establish forgetful functors, suspension functors, loop functors and Frobenius functors between such modules. The forgetful functors induce an inverse system of $\text{Ext}$ groups, and the inverse system stabilizes when the covariant module is bounded above. We define an analogue of the $\mathrm{\Lambda }$ algebra in this context and verify that its cohomology computes $\text{Ext}$.