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

Abstract

We show that the derived category of Brauer–Severi curves satisfies the Jordan–Hölder 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 ${rdim}_{cat}\left(X\right)=ind\left(X\right)-1$, 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.