Semiorthogonal decompositions for generalized Severi-Brauer schemes

Abstract: The purpose of this paper is to use conservative descent to study semi-orthogonal decompositions for some homogeneous varieties over general bases. We produce a semi-orthogonal decomposition for the bounded derived category of coherent sheaves on a generalized Severi-Brauer scheme. This extends known results for Sever-Brauer varieties and Grassmanianns. We use our results to construct semi-orthogonal decompositions for flag varieties over arbitrary bases. This generalises a result of Kapranov.