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

Abstract

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.