On the derived categories of coherent sheaves on some homogeneous spaces (Q1106313)

We quote in part from the paper: The derived category of coherent sheaves on the projective space P n has been shown to be equivalent (as a triangulated category) to the homotopy category of finite complexes of sheaves, consisting of finite direct sums of \({\mathcal O}(-i),\) \(i=0,1,...,n\). For a preadditive category \({\mathcal A}\) the author forms a triangulated category Tr(\({\mathcal A})\) which is ``generated freely'' by \({\mathcal A}\). Its objects are finite complexes, consisting of finite formal direct sums of objects of \({\mathcal A}\), and morphisms are homotopy classes of morphisms of complexes. The author then represents in the form Tr(\({\mathcal A})\) the derived categories of coherent sheaves on flag varieties and quadrics, and also the derived categories of finite- dimensional representations of parabolic subgroups of GL(n,\({\mathbb{C}})\).