Eilenberg Mac Lane spectra as p-cyclonic Thom spectra

Abstract

Hopkins and Mahowald gave a simple description of the mod $p$ Eilenberg Mac Lane spectrum ${\mathbb{F}}_p$ as the free ${\mathbb{E}}_2$-algebra with an equivalence of $p$ and $0$. We show for each faithful $2$-dimensional representation $\lambda$ of a $p$-group $G$ that the $G$-equivariant Eilenberg Mac Lane spectrum ${\underline{\mathbb{F}}}_p$ is the free ${\mathbb{E}}_{\lambda }$-algebra with an equivalence of $p$ and $0$. This unifies and simplifies recent work of Behrens, Hahn, and Wilson, and extends it to include the dihedral $2$-subgroups of O(2). The main new idea is that ${\underline{\mathbb{F}}}_p$ has a simple description as a $p$-cyclonic module over $\mathrm{THH}({\mathbb{F}}_{\mathrm{p}})$. We show our result is the best possible one in that it gives all groups $G$ and representations $V$ such that ${\underline{\mathbb{F}}}_p$ is the free ${\mathbb{E}}_V$-algebra with an equivalence of $p$ and $0$.

Revised in response to referee feedback. 19 pages