I have been reading the section in the beginning of *Fourier-Mukai and Nahm Transforms in Geometry and Mathematical Physics* by Bartocci et. al., and stumbled across the following sentence (page 26).

If we assume that $X$ is projective, since $G$ is finite there exists a $G$-equivariant ample line bundle, and this implies that every $G$-invariant sheaf has a left resolution by locally free $G$-equivariant sheaves.

Unfortunately the authors give no reference for this, is this an easy fact (I am still getting used to equivariant things)? Could some one provide a reference?

More generally, does this statement hold when $X$ is not projective? What about if $G$ is not finite?