On normed ${\mathbb{E}}_{\infty }$-rings in genuine equivariant $C_p$-spectra

Abstract

Genuine equivariant homotopy theory is equipped with a multitude of coherently commutative multiplication structures generalizing the classical notion of an ${\mathbb{E}}_{\infty }$-algebra. In this paper we study the $C_p$-${\mathbb{E}}_{\infty }$-algebras of Nardin--Shah with respect to a cyclic group $C_p$ of prime power order. We show that many of the higher coherences inherent to the definition of parametrized algebras collapse; in particular, they may be described more simply and conceptually in terms of ordinary ${\mathbb{E}}_{\infty }$-algebras as a diagram category which we call \emph{normed algebras}. Our main result provides a relatively straightforward criterion for identifying $C_p$-${\mathbb{E}}_{\infty }$-algebra structures. We visit some applications of our result to real motivic invariants.

37 pages, comments welcome