Eilenberg Mac Lane spectra as p$p$‐cyclonic Thom spectra

Abstract

Hopkins and Mahowald gave a simple description of the mod p $p$ Eilenberg Mac Lane spectrum F p ${\mathbb {F}}_p$ as the free E 2 $\mathbb {E}_2$ -algebra with an equivalence of p $p$ and 0. We show for each faithful 2-dimensional representation λ $\lambda$ of a p $p$ -group G $G$ that the G $G$ -equivariant Eilenberg Mac Lane spectrum F ̲ p $\underline{\mathbb {F}}_p$ is the free E λ $\mathbb {E}_{\lambda }$ -algebra with an equivalence of p $p$ 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 ) $\textnormal {O}(2)$ . The main new idea is that F ̲ p $\underline{\mathbb {F}}_p$ has a simple description as a p $p$ -cyclonic module over THH ( F p ) $\mathop {\rm THH}\nolimits (\mathbb {F}_p)$ . We show that our result is the best possible one in that it gives all groups G $G$ and representations V $V$ such that F ̲ p $\underline{\mathbb {F}}_p$ is the free E V $\mathbb {E}_V$ -algebra with an equivalence of p $p$ and 0.