A generalization of Serre's condition (F) with applications to the finiteness of unramified cohomology

TL;DR

This paper introduces a new field condition (F'_m) that generalizes Serre's condition (F) and ensures the finiteness of certain Galois cohomology groups, with broad applications.

Contribution

The paper defines the (F'_m) condition, extending Serre's (F), and demonstrates its implications for the finiteness of unramified cohomology and related structures.

Findings

(F'_m) condition implies finiteness of Galois cohomology groups.

Examples of fields satisfying (F'_m), including some not satisfying (F).

Applications to algebraic K-tori and étale cohomology groups.

Abstract

In this paper, we introduce a condition $\mathrm{(F}_m'\mathrm{)}$ on a field $K$, for a positive integer $m$, that generalizes Serre's condition (F) and which still implies the finiteness of the Galois cohomology of finite Galois modules annihilated by $m$ and algebraic $K$-tori that split over an extension of degree dividing $m$, as well as certain groups of \'etale and unramified cohomology. Various examples of fields satisfying $\mathrm{(F}_m'\mathrm{)}$, including those that do not satisfy (F), are given.