On residue maps for affine curves

TL;DR

This paper investigates compatibility of residue maps in étale and Galois cohomology for smooth affine algebraic curves with good reduction, aiding the study of unramified cohomology in number field function fields.

Contribution

It establishes key compatibility results between residue maps in étale and Galois cohomology for affine curves, advancing the understanding of their cohomological properties.

Findings

Proved compatibility results for residue maps in étale and Galois cohomology.

Applied these results to analyze finiteness of unramified cohomology.

Enhanced tools for studying cohomological properties of affine curves over number fields.

Abstract

We establish several compatibility results between residue maps in \'etale and Galois cohomology that arise naturally in the analysis of smooth affine algebraic curves having good reduction over discretely valued fields. These results are needed, and in fact have already been used, for the study of finiteness properties of the unramified cohomology of function fields of affine curves over number fields.