Semiorthogonal decompositions for generalised Severi-Brauer schemes

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