#### Volume 21, issue 7 (2021)

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Unstable modules with only the top $k$ Steenrod operations

### Zhulin Li

Algebraic & Geometric Topology 21 (2021) 3623–3662
##### 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 $\Lambda$ algebra in this context and verify that its cohomology computes $Ext$.

##### Keywords
Steenrod algebra, ringoid, top $k$ operations
Primary: 55S10
Secondary: 55U99