Unstable modules over the Steenrod algebra revisited

TL;DR

This paper provides a new, unified framework for understanding unstable modules over the Steenrod algebra by describing them as comodules over a bialgebra, extending previous work and exploring related categories at odd primes.

Contribution

It introduces a natural description of unstable modules as comodules over a bialgebra, unifying prior theories and extending to bigraded modules relevant to motivic cohomology.

Findings

Unified description of unstable modules as comodules over a bialgebra

Introduction of bigraded unstable modules related to motivic Steenrod algebra

Insights into the structure of unstable modules at odd primes

Abstract

A new and natural description of the category of unstable modules over the Steenrod algebra as a category of comodules over a bialgebra is given; the theory extends and unifies the work of Carlsson, Kuhn, Lannes, Miller, Schwartz, Zarati and others. Related categories of comodules are studied, which shed light upon the structure of the category of unstable modules at odd primes. In particular, a category of bigraded unstable modules is introduced; this is related to the study of modules over the motivic Steenrod algebra.