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 Ext groups, and the inverse system stabilizes when the covariant module is bounded above. We define an analogue of the Λ algebra in this context and verify that its cohomology computes Ext.