Remarks on chromatically localised hermitian K-theory

TL;DR

This paper explores chromatic localizations of hermitian K-theory spectra, extending known purity properties to Grothendieck-Witt theory and analyzing higher chromatic localizations.

Contribution

It generalizes the purity property of K(1)-local K-theory to Grothendieck-Witt theory and discusses higher chromatic localizations of L-theory spectra.

Findings

Purity property extends to Grothendieck-Witt theory.

Collected results on higher chromatic localizations.

Implications for L-theory spectra and Grothendieck-Witt theory.

Abstract

We describe chromatic localisations of genuine L-spectra of discrete rings and deduce that the purity property of $K(1)$-local $K$-theory of rings established by Bhatt-Clausen-Mathew also holds in Grothendieck-Witt theory. In addition, we collect some results on higher chromatic localisations of various L-theory spectra and their consequences for Grothendieck-Witt theory.