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As mentioned in the comments of this question, on a quasiprojective scheme over a field, every perfect complex is globally a complex of vector bundles.

I have some question about the extension of this to stacks:

  1. Is this true $G$ equivariantly? i.e. is it true for $X/G$ where $X$ is quasiprojective?
  2. As a slight extension of this, is this true for $\text{Bun}_G(X)$ where $X$ is a curve?

(and 3. it would be nice to have a modern reference to the original question about quasiprojective schemes, which I've not actually been able to find yet).

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    $\begingroup$ Yes, 1 is true, at least for bounded perfect complexes. The key point is that for every perfect complex of amplitude $[a,b]$, for a $G$-equivariant complex of vector bundles of amplitude $[a+1,b]$ and a $G$-equivariant morphism from that complex to the original complex that is a quasi-isomorphism on the "good truncations" $\tau_{>a}$, the cone is a perfect complex concentrated in one degree, which then has a $G$-equivariant homology sheaf that is a $G$-equivariant vector bundle. So now you can "adjoin" this to the $G$-equivariant complex of vector bundles. Surely there is a reference . . . $\endgroup$
    – Jason Starr
    Commented Aug 15, 2020 at 13:44

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