  It has recently been noticed that the degeneracies of the Poisson bracket of
linearized gravity on constant curvature Lorentzian manifold can be described
in terms of the cohomologies of a certain complex of differential operators.
This complex was first introduced by Calabi and its cohomology is known to be
isomorphic to that of the (locally constant) sheaf of Killing vectors. We
review the structure of the Calabi complex in a novel way, with explicit
calculations based on representation theory of GL(n), and also some tools for
studying its cohomology in terms of of locally constant sheaves. We also
conjecture how these tools would adapt to linearized gravity on other
backgrounds and to other gauge theories. The presentation includes explicit
formulas for the differential operators in the Calabi complex, arguments for
its local exactness, discussion of generalized Poincar\'e duality, methods of
computing the cohomology of locally constant sheaves, and example calculations
of Killing sheaf cohomologies of some black hole and cosmological Lorentzian
manifolds.
