A generalization of Serre’s condition $$\mathrm {(F)}$$ ( F ) with applications to the finiteness of unramified cohomology

Abstract

In this paper, we introduce a condition ( $$\mathrm {F}_m'$$ ) 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 ( $$\mathrm {F}_m'$$ ), including those that do not satisfy (F), are given.