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In this paper we show that the derived category of Brauer-Severi curvessatisfies the Jordan-H\"older property and cannot have quasi-phantoms, phantomsor universal phantoms. In this way we obtain that quasi-phantoms, phantoms oruniversal phantoms cannot exist in the derived category of smooth projectivecurves over a field $k$. Moreover, we show that a $n$-dimensional Brauer-Severivariety is completely characterized by the existence of a full weak exceptionalcollection consisting of pure vector bundles of length $n+1$, at least incharacteristic zero. We conjecture that Brauer-Severi varieties $X$ satisfy$\mathrm{rdim}_{\mathrm{cat}}(X)=\mathrm{ind}(X)-1$, provided period equalsindex, and prove this in the case of curves, surfaces and for Brauer-Severivarieties of index at most three. We believe that the results for curves areknown to the experts. We nevertheless give the proofs, adding to theliterature.

No phantoms in the derived category of curves over arbitrary fields, and derived characterizations of Brauer-Severi varieties