Norms and Hermitian $\mathrm{K}$-Theory

TL;DR

This paper develops a new algebro-geometric framework for hermitian K-theory using normed algebras in motivic spectra, demonstrating compatibility with the motivic infinite loop space machine.

Contribution

It introduces the theory of normed algebras in motivic spectra and shows that the motivic spectrum ko has a normed algebra structure compatible with orientations.

Findings

ko spectrum admits a normed algebra structure

Orientation map from MSL to ko respects the normed structure

Compatibility of the motivic infinite loop space machine with norms

Abstract

Over the past century, cohomology operations have played a crucial role in homotopy theory and its applications. A powerful framework for constructing such operations is the theory of commutative algebras in spectra. In this article, we discuss an algebro-geometric analogue of this framework, called the theory of normed algebras in motivic spectra. Specifically, we show that the motivic spectrum $\mathrm{ko}$ representing very effective hermitian $\mathrm{K}$-theory can be equipped with a normed algebra structure, and that the orientation map $\mathrm{MSL} \to \mathrm{ko}$ respects this structure. The main step will be showing that the motivic infinite loop space machine is compatible with norms.