The slice filtration and Grothendieck-Witt groups

TL;DR

This paper proves that the slice filtration on the Grothendieck-Witt group GW(k) over a perfect field corresponds exactly to the I-adic filtration, linking motivic homotopy theory with algebraic K-theory.

Contribution

It establishes that the slice filtration on GW(k) coincides with the I-adic filtration, providing a clear connection between motivic homotopy and algebraic structures.

Findings

Slice filtration on GW(k) equals I-adic filtration

Identifies the filtration in motivic stable homotopy theory

Connects motivic homotopy with algebraic K-theory

Abstract

Let k be a perfect field of characteristic different from two. We show that the filtration on the Grothendieck-Witt group GW(k) induced by the slice filtration for the sphere spectrum in the motivic stable homotopy category is the I-adic filtration, where I is the augmentation ideal in GW(k).