  We provide general guidelines for generalizing dynamical reduction models to
curved spacetimes and propose a class of generally covariant relativistic
versions of the GRW model. We anticipate that the collapse operators of our
class of models may play a r\^ole in a yet-to-be-formulated theory of
semiclassical gravity with collapses. We show explicitly that the collapse
operators map a dense domain of states that are initially Hadamard to final
Hadamard states -- a property that we expect will be needed for the
construction of such a semiclassical theory. Finally, we provide a simple
example in which we explicitly compute the violations in energy-momentum due to
the state reduction process and conclude that this violation is of the order of
a parameter of the model -- supposed to be small.
