On the $C_p$-equivariant dual Steenrod algebra

Abstract

We compute the $C_p$-equivariant dual Steenrod algebras associated to the constant Mackey functors ${\underline{\mathbb{F}}}_p$ and ${\underline{\mathbb{Z}}}_{(p)}$, as ${\underline{\mathbb{Z}}}_{(p)}$-modules. The $C_p$-spectrum ${\underline{\mathbb{F}}}_p\otimes {\underline{\mathbb{F}}}_p$ is not a direct sum of $RO(C_p)$-graded suspensions of ${\underline{\mathbb{F}}}_p$ when $p$ is odd, in contrast with the classical and $C_2$-equivariant dual Steenrod algebras.

15 pages, comments welcome!