On equivariant derived categories

We study the equivariant category associated to a finite group action on the derived category of coherent sheaves of a smooth projective variety. In particular, we discuss decompositions of the equivariant category, prove the existence of a Serre functor, and give a criterion for the equivariant category to be Calabi–Yau. We describe an obstruction for a subgroup of the group of auto-equivalences to act on the derived category. As application we show that the equivariant category of any Calabi–Yau action on the derived category of an elliptic curve is equivalent to the derived category of an elliptic curve.