TY - JOUR

T1 - Milnor K-Theory of p-Adic rings

AU - Lüders, Morten

AU - Morrow, Matthew

N1 - Funding Information: The first author was supported by the DFG Research Fellowship LU 2418/1-1. The second author was partly supported by the ANR JCJC project Périodes en Géométrie Arithmétique et Motivique (ANR-18-CE40-0017). Acknowledgment: We thank Bhargav Bhatt, Dustin Clausen, and Akhil Mathew for discussions and Shuji Saito for related correspondence. We are grateful to Elden Elmanto for comments on the paper, and to the anonymous referees for various suggestions and improvements.

PY - 2023/3/1

Y1 - 2023/3/1

N2 - We study the mod pr 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 pr 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.

AB - We study the mod pr 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 pr 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.

KW - math.KT

UR - http://www.scopus.com/inward/record.url?scp=85144012998&partnerID=8YFLogxK

U2 - 10.48550/arXiv.2101.01092

DO - 10.48550/arXiv.2101.01092

M3 - Article

AN - SCOPUS:85144012998

VL - 2023

SP - 69

EP - 116

JO - Journal fur die Reine und Angewandte Mathematik

JF - Journal fur die Reine und Angewandte Mathematik

SN - 0075-4102

IS - 796

ER -