  We give the detailed derivation of the fully covariant form of the quadratic
action and the derived linear equations of motion for a massive graviton in an
arbitrary background metric (which were presented in arXiv:1410.8302 [hep-th]).
Our starting point is the de Rham-Gabadadze-Tolley (dRGT) family of ghost free
massive gravities and using a simple model of this family, we are able to
express this action and these equations of motion in terms of a single metric
in which the graviton propagates, hence removing in particular the need for a
"reference metric" which is present in the non perturbative formulation. We
show further how 5 covariant constraints can be obtained including one which
leads to the tracelessness of the graviton on flat space-time and removes the
Boulware-Deser ghost. This last constraint involves powers and combinations of
the curvature of the background metric. The 5 constraints are obtained for a
background metric which is unconstrained, i.e. which does not have to obey the
background field equations. We then apply these results to the case of Einstein
space-times, where we show that the 5 constraints become trivial, and
Friedmann-Lema\^{\i}tre-Robertson-Walker space-times, for which we correct in
particular some results that appeared elsewhere. To reach our results, we
derive several non trivial identities, syzygies, involving the graviton fields,
its derivatives and the background metric curvature. These identities have
their own interest. We also discover that there exist backgrounds for which the
dRGT equations cannot be unambiguously linearized.
