The Homogeneous Spectrum of Milnor-Witt $K$-Theory

TL;DR

This paper determines the structure of homogeneous prime ideals in Milnor-Witt K-theory over fields, linking it to classical Witt ring results and advancing the understanding of tensor triangular geometry in motivic homotopy theory.

Contribution

It explicitly computes the Zariski spectrum of homogeneous prime ideals in Milnor-Witt K-theory, connecting it to classical results and supporting Balmer's program in motivic homotopy theory.

Findings

Determined the Zariski spectrum of homogeneous prime ideals in Milnor-Witt K-theory.

Revealed connections between Milnor-Witt K-theory and classical Witt ring results.

Laid groundwork for tensor triangular geometry in stable motivic homotopy category.

Abstract

For any field $F$ (of characteristic not equal to 2), we determine the Zariski spectrum of homogeneous prime ideals in $K^{MW}_*(F)$, the Milnor-Witt $K$-theory ring of $F$. As a corollary, we recover Lorenz and Leicht's classical result on prime ideals in the Witt ring of $F$. Our computation can be seen as a first step in Balmer's program for studying the tensor triangular geometry of the stable motivic homotopy category.