Some Remarks on the Derived Categories of Coherent Sheaves on Homogeneous Spaces

In this paper we prove first a general theorem on semiorthogonal decompositions in derived categories of coherent sheaves for flat families over a smooth base. We then show that the derived categories of coherent sheaves on flag varieties of classical type are generated by complete exceptional collections. Finally, we find complete exceptional collections in the derived categories of some homogeneous spaces of the symplectic groups of small rank.