A remark on Gersten complex for Milnor $K$-theory

TL;DR

This paper proves the exactness of the Gersten complex for Milnor K-theory over certain Henselian local schemes in degrees above the dimension, advancing understanding of algebraic K-theory structures.

Contribution

It establishes the exactness of the Gersten complex for Milnor K-theory over essentially smooth Henselian local schemes in degrees at least the dimension, which was previously unknown.

Findings

Gersten complex is exact in degrees ≥ dimension for these schemes

Results apply to regular local Henselian domains

Enhances understanding of Milnor K-theory in algebraic geometry

Abstract

In this note, we consider the Gersten complex for Milnor $K$-theory over a regular local Henselian domain $S$ and prove that in degrees $\geq \dim S\geq 1$, the Gersten complex of an essentially smooth Henselian local $S$-scheme is exact.