For any finite subgroup G ⊂ SL3(ℂ), work of Bridgeland–King–Reid constructs an equivalence between the G-equivariant derived category of ℂ3 and the derived category of the crepant resolution Y = G-Hilb(ℂ3) of ℂ3/G. When G is abelian, we show that this equivalence gives a natural correspondence between irreducible representations of G and exceptional subvarieties of Y, thereby extending the McKay correspondence from two to three dimensions. This categorifies Reid's recipe and extends earlier work from [J. reine angew. Math. 636 (2009), 193–236] and [J. Algebra 324 (2010), no. 8, 2064–2087] which dealt only with the case when ℂ3/G has one isolated singularity.
Cautis, S., Craw, A., & Logvinenko, T. (2017). Derived Reid's recipe for abelian subgroups of SL3(ℂ). Journal Fur Die Reine Und Angewandte Mathematik, 2017(727), 1-48. https://doi.org/10.1515/crelle-2014-0086