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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) andwhich still implies the finiteness of the Galois cohomology of finite Galoismodules annihilated by $m$ and algebraic $K$-tori that split over an extensionof degree dividing $m$, as well as certain groups of \'etale and unramifiedcohomology. Various examples of fields satisfying $\mathrm{(F}_m'\mathrm{)}$,including those that do not satisfy (F), are given.

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