  It known from the work of Feigin-Tsygan, Weibel and Keller that the
cohomology groups of a smooth complex variety X can be recovered from (roughly
speaking) its derived category of coherent sheaves. In this paper we show that
for a finite group G acting on X the same procedure applied to G-equivariant
sheaves gives the orbifold cohomology of X/G.
  As an application, in some cases we are able to obtain simple proofs of an
additive isomorphism between the orbifold cohomology of X/G and the usual
cohomology of its crepant resolution (the general equality of Euler and Hodge
numbers was obtained earlier by various authors). We also state some
conjectures on the product structures, as well as the singular case; and a
connection with recent work by Kawamata.
