  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.
