Witt sheaves and the $\eta$-inverted sphere spectrum

TL;DR

This paper explores the motivic analog of Serre's finiteness theorem using Witt sheaves and the $ a$-inverted sphere spectrum, establishing torsion properties and a Witt motives category over fields.

Contribution

It introduces a motivic framework for Witt theory, computes stable operations, and constructs a Witt motives category, linking it to the minus part of the rational stable homotopy category.

Findings

$ a$-inverted sphere spectrum's homotopy groups are torsion for positive degrees

A category of Witt motives over a field is defined and shown to be equivalent to the minus part of $SH(k)_Q$

Computations of stable operations and cooperations of rational Witt theory enable these results.

Abstract

Ananyevsky has recently computed the stable operations and cooperations of rational Witt theory. These computations enable us to show a motivic analog of Serre's finiteness result: Theorem: Let $k$ be a field. Then $\pi^{\mathbb{A}^1}_ n(\mathbb{S}^- _k )_*$ is torsion for $n > 0$. As an application we define a category of Witt motives over $k$ and show that rationally this category is equivalent to the minus part of $SH(k)_\mathbb{Q}$.