  Let X and Y be two smooth Deligne-Mumford stacks and consider a function f,
resp. g, on X, resp. Y. Assume that there exists a complex F of sheaves on the
fiber product of X and Y over A^1 (induced by f and g), such that the
Fourier-Mukai transform with the kernel F gives an equivalence between the
bounded derived categories of coherent sheaves on X and Y. If X_0 Y_0 are the
fibers of f and g over zero, respectively, we show that the singular derived
categories of X_0 and Y_0 are also equivalent. We apply this statement in the
setting of McKay correspondence, and generalize a result of Orlov on the
derived category of a Calabi-Yau hypersurface in a weighted projective space,
to products of Calabi-Yau hypersurfaces in simplicial toric varieties with nef
anticanonical class.
