  Let g=g_{0} \oplus g_{1} be a classical Lie superalgebra and F be the
category of finite dimensional g-supermodules which are semisimple over g_{0}.
In this paper we investigate the homological properties of the category F. In
particular we prove that F is self-injective in the sense that all projective
supermodules are injective. We also show that all supermodules in F admit a
projective resolution with polynomial rate of growth and, hence, one can study
complexity in F. If g is a Type I Lie superalgebra we introduce support
varieties which detect projectivity and are related to the associated varieties
of Duflo and Serganova. If in addition g has a (strong) duality then we prove
that the conditions of being tilting or projective are equivalent.
