Unstable Modules with the Top $k$ Squares

Unstable modules over the Steenrod algebra with only the top $k$ operations are introduced in the language of ringoids. We prove the category of such modules has homological dimension at most $k$. A pratical method, which generalizes the $\Lambda$ complex, to compute the $\mathrm{Ext}$ group from such modules to spheres is proposed. We are also able to establish several functors to relate such modules and unstable modules over the Steenrod algebra, and to describe the connections between the $\mathrm{Ext}$ groups in them.