  The Coleman-Mandula (CM) theorem states that the Poincar\'e and internal
symmetries of a Minkowski spacetime quantum field theory cannot combine
nontrivially in an extended symmetry group. We establish an analogous result
for quantum field theory in curved spacetimes, assuming local covariance, the
timeslice property, a local dynamical form of Lorentz invariance, and
additivity. Unlike the CM theorem, our result is valid in dimensions $n\ge 2$
and for free or interacting theories. It is formulated for theories defined on
a category of all globally hyperbolic spacetimes equipped with a global
coframe, on which the restricted Lorentz group acts, and makes use of a general
analysis of symmetries induced by the action of a group $G$ on the category of
spacetimes. Such symmetries are shown to be canonically associated with a
cohomology class in the second degree nonabelian cohomology of $G$ with
coefficients in the global gauge group of the theory. Our main result proves
that the cohomology class is trivial if $G$ is the universal cover $\cal S$ of
the restricted Lorentz group. Among other consequences, it follows that the
extended symmetry group is a direct product of the global gauge group and $\cal
S$, all fields transform in multiplets of $\cal S$, fields of different spin do
not mix under the extended group, and the occurrence of noninteger spin is
controlled by the centre of the global gauge group. The general analysis is
also applied to rigid scale covariance.
