# Vanishing lines for modules over the motivic Steenrod algebra

## TL;DR

This paper investigates conditions under which modules over specific subalgebras of the motivic Steenrod algebra exhibit freeness and possess a vanishing line, using Margolis homology to establish these criteria.

## Contribution

It introduces new criteria based on Margolis homology for determining freeness and vanishing lines in modules over motivic Steenrod algebra subalgebras.

## Key findings

- Criteria for freeness of modules established
- Vanishing lines characterized via Margolis homology
- Results applicable to modules over motivic Steenrod algebra at prime 2

## Abstract

We study criteria for freeness and for the existence of a vanishing line for modules over certain Hopf subalgebras of the motivic Steenrod algebra over $\mathrm{Spec}(\mathbb{C})$ at the prime 2. These turn out to be determined by the vanishing of certain Margolis homology groups in the quotient Hopf algebra $\mathcal{A}/\tau$.