On the Steenrod module structure of $\mathbb{R}$-motivic Spanier-Whitehead duals

TL;DR

This paper investigates the structure of $ ext{R}$-motivic cohomology of duals of finite complexes, providing a method to determine their module structure over the $ ext{R}$-motivic Steenrod algebra, with applications to self-duality of certain modules.

Contribution

It introduces a method to recover the $ ext{R}$-motivic cohomology of dual spectra as modules over the Steenrod algebra, based on the cohomology of the original spectrum, and applies this to classify self-dual module structures.

Findings

16 out of 128 module structures on $ ext{A}^{ ext{R}}(1)$ are self-dual.

A procedure to determine the dual module structure from the original.

Characterization of self-duality in $ ext{A}^{ ext{R}}$-module structures.

Abstract

The $\mathbb{R}$-motivic cohomology of an $\mathbb{R}$-motivic spectrum is a module over the $\mathbb{R}$-motivic Steenrod algebra $\mathcal{A}^{\mathbb{R}}$. In this paper, we describe how to recover the $\mathbb{R}$-motivic cohomology of the Spanier-Whitehead dual $\mathrm{DX}$ of an $\mathbb{R}$-motivic finite complex $\mathrm{X}$, as an $\mathcal{A}^{\mathbb{R}}$-module, given the $\mathcal{A}^{\mathbb{R}}$-module structure on the cohomology of $\mathrm{X}$. As an application, we show that 16 out of 128 different $\mathcal{A}^{\mathbb{R}}$-module structures on $\mathcal{A}^{\mathbb{R}}(1):= \langle \mathrm{Sq}^1, \mathrm{Sq}^2 \rangle$ are self-dual.