  For a field theory that is invariant under diffeomorphisms there is a subtle
interplay between symmetries, conservation laws and the phase space of the
theory. The natural language for describing these ideas is that of differential
forms and both differential forms on space-time and differential forms on the
infinite dimensional space of solutions of the equations of motion of the field
theory play an important role. There are exterior derivatives on both spaces
and together they weave a double differential complex which captures the
cohomology of the theory. This is important in the definition of invariants in
general relativity, such as mass and angular momentum and is also relevant to
the study of quantum anomalies in gauge theories.
  We derive the structure of this double complex and show how it relates to
conserved quantities in gravitational theories. One consequence of the
construction is that conserved quantities can be calculated exactly at finite
distance --- for example it is not necessary to go to asymptotic regimes to
calculate the mass or angular momentum of a stationary solution of Einstein's
equations, they can be obtained exactly by an integration over any sphere
outside the mass even at finite radius.
