Milnor-Witt $K$-groups of local rings

TL;DR

This paper defines Milnor-Witt K-groups for local rings, relating them to Witt rings and Milnor K-theory, extending previous field-based results to local rings with infinite fields.

Contribution

It introduces Milnor-Witt K-groups for local rings and establishes their structure as pull-backs involving Witt rings and Milnor K-groups, generalizing known results from fields.

Findings

Milnor-Witt K-groups of local rings are characterized as pull-backs.

The structure relates to the fundamental ideal in Witt rings.

Generalizes Milnor-Witt K-group results from fields to local rings.

Abstract

We introduce Milnor-Witt $K$-groups of local rings and show that the $n$th Milnor-Witt $K$-group of a local ring $R$ which contains an infinite field of characteristic not $2$ is the pull-back of the $n$th power of the fundamental ideal in the Witt ring of $R$ and the $n$th Milnor $K$-group of $R$ over the $n$th Milnor $K$-group of $R$ modulo $2$. This generalizes the work of Morel-Hopkins on Milnor-Witt $K$-groups of fields.