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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?

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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.

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  • $\begingroup$ I see that finiteness is heavily used, but since a finite resolution of locally free sheaves exists on any regular variety, I’m not sure that projectivity hypotheses are needed. $\endgroup$ – DKS Feb 6 at 2:15
  • $\begingroup$ Also, thank you for enlightening me with regards to the claimed fact. $\endgroup$ – DKS Feb 6 at 2:18
  • $\begingroup$ @DKS Ah, I didn't know that fact. It does seem then that you could use averaging to construct an equivariant resolution. Do you have a reference? $\endgroup$ – Phil Tosteson Feb 6 at 5:13
  • $\begingroup$ I did end up finding a discussion in Chriss and Ginsburg, Representation Theory and Complex Geometry, the beginning of chapter 5. $\endgroup$ – DKS Feb 6 at 11:05
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    $\begingroup$ A reference for my comment is an exercise in Hartshorne, number III.6.9a if I remember correctly. $\endgroup$ – DKS Feb 6 at 11:05

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