We consider the problem of comparing -structures under the derived McKay correspondence and for tilting equivalences. In low-dimensional cases, we relate the -structures via torsion theories arising from additive functions on the triangulated category. As an application, we give a criterion for rationality for surfaces with a tilting bundle. We also show that every smooth projective surface that admits a full, strong, and exceptional collection of line bundles is rational.
"The McKay Correspondence, Tilting, and Rationality." Michigan Math. J. 66 (4) 785 - 811, November 2017. https://doi.org/10.1307/mmj/1501034511