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

2017/01/11 by Novaković, Saša
#14F05 #17C20 #18E30 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1701.03020

Abstract

In this paper 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(X)=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.

Related

• No full exceptional collections on non-split Brauer--Severi varieties of dimension ≤ 3
• Algebraic cycles on Severi-Brauer schemes of prime degree over a curve
• Rational points on symmetric powers and categorical representability
• A Looming of phantoms
• Construction of Brauer-Severi Varieties