Milnor K-theory of complete discrete valuation rings with finite residue fields

TL;DR

This paper proves that for complete discrete valuation rings with finite residue fields, the improved Milnor K-theory map is an isomorphism in degrees ≥ 3, confirming the Gersten conjecture in this setting, including the p-adic case.

Contribution

It establishes the isomorphism of improved Milnor K-theory maps in degrees ≥ 3 for such rings, confirming the Gersten conjecture in this context, especially for p-adic rings.

Findings

The map on improved Milnor K-theory is an isomorphism in degrees ≥ 3.

The result confirms the Gersten conjecture for these rings.

The finding is new in the p-adic case.

Abstract

Consider a complete discrete valuation ring $\mathcal{O}$ with quotient field $F$ and finite residue field. Then the inclusion map $\mathcal{O} \hookrightarrow F$ induces a map $\hat{\mathrm{K}}^\mathrm{M}_*\mathcal{O} \to \hat{\mathrm{K}}^\mathrm{M}_*F$ on improved Milnor K-theory. We show that this map is an isomorphism in degrees bigger or equal to 3. This implies the Gersten conjecture for improved Milnor K-theory. This result is new in the $p$-adic case.