On the relative Gersten conjecture for Milnor K-theory in the smooth case

TL;DR

This paper investigates the exactness properties of the Gersten complex for Milnor K-sheaves on smooth schemes over discrete valuation rings, providing new insights into its behavior and conditions for exactness.

Contribution

It demonstrates the near-exactness of the Gersten complex for Milnor K-sheaves in the smooth case and links the exactness at the first place to the associated discrete valuation ring.

Findings

Gersten complex is exact except at the first position

Exactness at the first position can be checked at the generic point's discrete valuation ring

Complements previous results for K-theory, motivic cohomology, and étale cohomology

Abstract

We show that the Gersten complex for the (improved) Milnor K-sheaf on a smooth scheme over an excellent discrete valuation ring is exact except at the first place and that exactness at the first place may be checked at the discrete valuation ring associated to the the generic point of the special fiber. This complements results of Gillet and Levine for K-theory, Geisser for motivic cohomology and Schmidt and Strunk and the author for \'etale cohomology.