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

Question

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?

Answer 1

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.