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The derived McKay correspondence as proved by Bridgeland, King and Reid was considered for algebraically closed fields of characteristic zero. Does this correspondence hold over a field $k$ of characteristic zero without assuming it is algebraically closed? There is a thesis of Blume, "McKay correspondence and G-Hilbert schemes", where he proved the classical correspondence for fields that are not algebraically closed. What is about the derived version in the sense of Bridgland, King and Reid?

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I think it should hold over a field $k$ if the crepant resolution is defined over this field and if the kernel giving the Fourier--Mukai functor is defined over this field as well. Indeed, to check that the functor is an equivalence you can go to the closure of the field where it holds. So, if the resolution is given by a Hilbert scheme and the kernel is given by the universal subscheme, everything is defined over $k$ and so the derived McKay should hold.

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