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For any finite subgroupG  ⊂    SL  3   ⁡   (  ℂ  ){G\subset\operatorname{SL}_{3}(\mathbb{C})}   , work of Bridgeland–King–Reid constructs an equivalence between the  G -equivariant derived category ofℂ  3    {\mathbb{C}^{3}}    and the derived category of the crepant resolutionY  =    G  ⁢  -  ⁢  Hilb    ℂ  3{Y=\mathop{{G}\hbox{-}\mathrm{Hilb}}\mathbb{C}^{3}}    ofℂ  3   /  G    {\mathbb{C}^{3}/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 [ 3 ] and [ 14 ] which dealt only with the case whenℂ  3   /  G    {\mathbb{C}^{3}/G}    has one isolated singularity.

Derived Reid's recipe for abelian subgroups of SL 3 (ℂ)