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The theorem of Bridgeland-King-Reid says that if $M$ is a smooth quasi-projective complex variety of dimension at most $3$ on which a finite group $G$ acts such that the canonical sheaf $\omega_M$ is $G$-equivariantly locally trivial, then $M/G$ has a crepant resolution, $Y$, such that $D^b(Y)\cong D^b_G(M)$. In particular, $D^b_G(M)$ inherits from $D^b(Y)$ a new (?) tensor structure. Does this tensor product have a nice interpretation in terms of complexes of $G$-equivariant sheaves?

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    $\begingroup$ the same could be asked about any derived equivalence and it seems a very interesting (but also very hard!) question. As far as I'm aware, Tarig Abdel Gadir, a former student of Alastair Craw, has done some work on this, although a pdf has yet to appear on the arxiv. $\endgroup$ – Jacob Bell Mar 11 '13 at 13:27
  • $\begingroup$ Dimension equals three? $\endgroup$ – Jason Starr Mar 12 '13 at 15:21
  • $\begingroup$ @Jason: exactly. I corrected the original question. $\endgroup$ – Benjamin Antieau Mar 12 '13 at 18:44

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