  We study the images of tautological bundles on Hilbert schemes of points on
surfaces and their wedge powers under the derived McKay correspondence. The
main observation of the paper is that using a derived equivalence differing
slightly from the standard one considerably simplifies both the results and
their proofs. As an application, we obtain shorter proofs for known results as
well as new formulae for homological invariants of tautological sheaves. In
particular, we compute the extension groups between wedge powers of
tautological bundles associated to line bundles on the surface.
