Total power operations in spectral sequences

TL;DR

This paper explores how power operations behave in spectral sequences, specifically in equivariant and localized contexts, providing new computational tools and insights into equivariant stable homotopy theory.

Contribution

It introduces a method for descending power operations through homotopy limit spectral sequences and applies it to compute norms and power operations in equivariant and localized spectra.

Findings

Describes descent of power operations in spectral sequences

Computes norms in the $C_2$-equivariant Adams spectral sequence

Calculates power operations for the $K(1)$-local sphere

Abstract

We describe how power operations descend through homotopy limit spectral sequences. We apply this to describe how norms appear in the $C_2$-equivariant Adams spectral sequence, to compute norms on $\pi_0$ of the equivariant $KU$-local sphere, and to compute power operations for the $K(1)$-local sphere. An appendix contains material on equivariant Bousfield localizations which may be of independent interest.