# Resolution by locally free $G$-equivariant sheaves on varieties

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 fi nite 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?

Suppose $$L$$ on $$X$$ is a (very) ample line bundle. Then $$\oplus_{g \in G} g^* L$$ is a $$G$$ equivariant vector bundle. Its determinant is also $$G$$ equivariant and isomorphic to $$\otimes_{g \in G} {g^*L}$$, with each factor (very) ample. Thus it is (very) ample.
Now this line bundle gives us a $$G$$-equivariant embedding into projective space, and to construct the resolution by $$G$$-equivariant vector bundles we can construct a resolution using bundles of the form $$\oplus_{g \in G} g^* \mathcal O(n)$$. Other than the "averaging trick," the construction of this resolution is identical to Serre's non-equivariant resolution, i.e. twist high enough that your sheaf is globally generated, use a surjection from a trivial bundle, twist back down, and then repeat for the kernel.
I've used the finiteness of $$G$$ and the projectivity of $$X$$ heavily, and I'm not sure what you can say without them.