Equivariant chromatic localizations and commutativity

TL;DR

This paper investigates how certain localizations in equivariant stable homotopy theory affect the preservation of structured multiplications, establishing that localizations with trivial actions preserve equivariant commutative ring spectra.

Contribution

It provides new results on the preservation of algebraic structures under localizations in equivariant spectra, including a general proof for trivial action localizations.

Findings

Bousfield localization with trivial action preserves equivariant commutative ring spectra

Analysis of acyclic categories in equivariant spectra

Results on the preservation of structured multiplications

Abstract

In this paper, we study the extent to which Bousfield and finite localizations relative to a thick subcategory of equivariant finite spectra preserve various kinds of highly structured multiplications. Along the way, we describe some basic, useful results for analyzing categories of acyclics in equivariant spectra, and we show that Bousfield localization with respect to an ordinary spectrum (viewed as an equivariant spectrum with trivial action) always preserves equivariant commutative ring spectra.