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