On the $C_p$-equivariant dual Steenrod algebra

TL;DR

This paper computes the $C_p$-equivariant dual Steenrod algebras for certain Mackey functors, revealing differences from classical cases especially at odd primes, and enhances understanding of equivariant stable homotopy theory.

Contribution

It provides explicit calculations of the $C_p$-equivariant dual Steenrod algebras for constant Mackey functors, highlighting novel structural differences at odd primes.

Findings

The $C_p$-spectrum $_p _p$ is not a direct sum of $RO(C_p)$-graded suspensions when $p$ is odd.

The structure of the $C_p$-equivariant dual Steenrod algebra differs from classical and $C_2$ cases at odd primes.

The work advances the understanding of equivariant cohomology operations for cyclic groups.

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.