Unstable Modules with the Top $k$ Squares

TL;DR

This paper introduces a new category of unstable modules over the Steenrod algebra limited to the top k operations, establishing their homological properties and providing computational tools for Ext groups.

Contribution

It defines unstable modules with only the top k Steenrod operations, proves their homological dimension bounds, and generalizes the $ ext{Lambda}$ complex for Ext computations.

Findings

Category of such modules has homological dimension at most k

A generalized $ ext{Lambda}$ complex method for Ext computation

Connections established between Ext groups in different module categories

Abstract

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.