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

Brian Shin

TL;DR

This work develops normed algebras in motivic spectra, a richer framework than ordinary commutative algebras, to study multiplicative structures in motivic homotopy theory. The authors prove that the motivic spectrum $ko_S$ representing very effective hermitian K-theory admits a normed algebra structure and that the orientation $MSL o ko_S$ is compatible with norms, using a motivic infinite loop space machine compatible with norm operations. They place this in a broader program by showing how finite étale descent yields normed enhancements for several motivic spectra ($ ext{H}b{Z}_S$, $ ext{KGL}_S$, $ ext{MGL}_S$, $ ext{MSL}_S$) and how framed transfers interact with normed structures via the motivic recognition principle, culminating in a proof strategy that also yields a normed structure on Milnor–Witt motivic cohomology $ ext{H} ilde{b{Z}}_S$ with a normed map from $ko_S$. The work highlights subtle obstacles, such as the nontrivial behavior under Bott inversion, and clarifies how norms interact with transfers, cobordism, and K-theory in the motivic setting, advancing the understanding of power operations in motivic contexts.

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.

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

TL;DR

This work develops normed algebras in motivic spectra, a richer framework than ordinary commutative algebras, to study multiplicative structures in motivic homotopy theory. The authors prove that the motivic spectrum representing very effective hermitian K-theory admits a normed algebra structure and that the orientation is compatible with norms, using a motivic infinite loop space machine compatible with norm operations. They place this in a broader program by showing how finite étale descent yields normed enhancements for several motivic spectra (, , , ) and how framed transfers interact with normed structures via the motivic recognition principle, culminating in a proof strategy that also yields a normed structure on Milnor–Witt motivic cohomology with a normed map from . The work highlights subtle obstacles, such as the nontrivial behavior under Bott inversion, and clarifies how norms interact with transfers, cobordism, and K-theory in the motivic setting, advancing the understanding of power operations in motivic contexts.

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 representing very effective hermitian -theory can be equipped with a normed algebra structure, and that the orientation map respects this structure. The main step will be showing that the motivic infinite loop space machine is compatible with norms.
Paper Structure (6 sections, 10 theorems, 24 equations, 1 table)

This paper contains 6 sections, 10 theorems, 24 equations, 1 table.

Table of Contents

  1. Introduction
  2. A Brief Tour of Motivic Homotopy Theory
  3. Commutative Monoids and Commutative Algebras via Correspondences
  4. Norms in Motivic Homotopy Theory
  5. Motivic Infinite Loop Space Theory
  6. Norms and Transfers, K-theory and Cobordism

Key Result

Theorem 1.1

Let $S$ be a quasi-compact quasi-separated scheme. The motivic spectrum $\mathrm{ko}_S \in \mathscr{SH}(S)$ admits a normed algebra structure. In fact, the diagram \begin{tikzcd} \mathrm{MSL}_S \ar[d] \ar[r] & \mathrm{ko}_S \ar[d] \\ \mathrm{MGL}_S \ar[r] & \mathrm{kgl}_S

Theorems & Definitions (48)

  • Theorem 1.1: Shin_NormsTransfers
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 2.6
  • Definition 2.7
  • Example 2.8
  • Example 2.9
  • Remark 2.10
  • Remark 2.11
  • ...and 38 more