Milnor $K$-groups and function fields of hypersurfaces in positive characteristic

TL;DR

This paper explores the relationship between Milnor K-groups and function fields of hypersurfaces in positive characteristic, building on recent results about Kähler differentials and employing the Bloch-Gabber-Kato theorem.

Contribution

It investigates an analogous description of the kernel of the restriction homomorphism in mod-$p$ Milnor K-theory for hypersurfaces over fields of characteristic p.

Findings

Provides insights into the kernel of the restriction homomorphism in Milnor K-theory

Connects K-theory with differential forms via Bloch-Gabber-Kato theorem

Extends previous results on Kähler differentials to Milnor K-groups

Abstract

Let $X$ be an integral affine or projective hypersurface over a field $F$ of characteristic $p>0$, and let $F(X)$ denote its function field. In a recent article, Dolphin and Hoffmann obtained an explicit description of the kernel of the natural restriction homomorphism between the rings of absolute K\"{a}hler differentials of $F$ and $F(X)$, respectively. In this note, we examine the possibility of deriving an analogous result for mod-$p$ Milnor $K$-theory using the Bloch-Gabber-Kato theorem.