Milnor $K$-theory of $p$-adic rings

TL;DR

This paper investigates the mod p^r Milnor K-groups of p-adically complete rings, establishing new descriptions via syntomic cohomology and proving conjectures related to Gersten and Bloch-Kato-Gabber theorems in specific algebraic contexts.

Contribution

It provides a Nesterenko-Suslin style description of mod p^r Milnor K-groups and proves the mod p^r Gersten conjecture for Milnor K-theory over certain schemes, extending known theorems to valuation rings and formal schemes.

Findings

Established a description of Milnor K-groups in terms of syntomic cohomology.

Proved the mod p^r Gersten conjecture for smooth schemes over complete DVRs.

Extended Bloch-Kato-Gabber theorem validity to valuation rings and formal schemes.

Abstract

We study the mod $p^r$ Milnor $K$-groups of $p$-adically complete and $p$-henselian rings, establishing in particular a Nesterenko-Suslin style description in terms of the Milnor range of syntomic cohomology. In the case of smooth schemes over complete discrete valuation rings we prove the mod $p^r$ Gersten conjecture for Milnor $K$-theory locally in the Nisnevich topology. In characteristic $p$ we show that the Bloch-Kato-Gabber theorem remains true for valuation rings, and for regular formal schemes in a pro sense.