Class field theory, Hasse principles and Picard-Brauer duality for two-dimensional local rings

TL;DR

This paper explores the arithmetic duality for two-dimensional local rings, deriving class field theory, Hasse principles, and a duality between divisor class groups and Brauer groups through analysis of cohomology structures.

Contribution

It establishes concrete consequences of arithmetic duality for two-dimensional local rings, including class field theory and Hasse principles, with new insights into cohomology group structures.

Findings

Derived class field theory for 2D local rings

Established Hasse principles for coverings and K2

Proved finiteness properties of cohomology groups

Abstract

We draw concrete consequences from our arithmetic duality for two-dimensional local rings with perfect residue field. These consequences include class field theory, Hasse principles for coverings and $K_{2}$ and a duality between divisor class groups and Brauer groups. To obtain these, we analyze the ind-pro-algebraic group structures on arithmetic cohomology obtained earlier and prove some finiteness properties about them.