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

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.