Smashing Localizations in Equivariant Stable Homotopy

TL;DR

This paper investigates the behavior of smashing Bousfield localizations in equivariant stable homotopy theory, establishing key theorems and extensions for various equivariant Johnson-Wilson theories and their algebraic structures.

Contribution

It demonstrates that smash product and chromatic convergence theorems hold for equivariant theories only after Borel completion and explores how localizations affect norms in $N_ abla$-algebras.

Findings

Chromatic convergence theorems hold after Borel completion for $E_{ r{R}}(n)$.

Analogous results are established for $C_{2^n}$-equivariant Johnson-Wilson theories.

Induced localizations enhance the norms available for $N_ abla$-algebras.

Abstract

We study how smashing Bousfield localizations behave under various equivariant functors. We show that the analogs of the smash product and chromatic convergence theorems for the Real Johnson-Wilson theories $E_{\mathbb{R}}(n)$ hold only after Borel completion. We establish analogous results for the $C_{2^n}$-equivariant Johnson-Wilson theories constructed by Beaudry, Hill, Shi, and Zeng. We show that induced localizations upgrade the available norms for an $N_\infty$-algebra, and we determine which new norms appear. Finally, we explore generalizations of our results on smashing localizations in the context of a quasi-Galois extension of $E_\infty$-rings.