Decompositions of derived categories of gerbes and of families of Brauer-Severi 
varieties

Bergh, Daniel; Schnürer, Olaf M.

doi:10.25537/dm.2021v26.1465-1500

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Decompositions of derived categories of gerbes and of families of Brauer-Severi varieties

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Bergh, Daniel
Max Planck Institute for Mathematics, Max Planck Society;

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https://doi.org/10.25537/dm.2021v26.1465-1500
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https://doi.org/10.48550/arXiv.1901.08945
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Citation

Bergh, D., & Schnürer, O. M. (2021). Decompositions of derived categories of gerbes and of families of Brauer-Severi varieties. Documenta Mathematica, 26, 1465-1500. doi:10.25537/dm.2021v26.1465-1500.

Cite as: https://hdl.handle.net/21.11116/0000-000A-D2EE-D

Abstract

It is well known that the category of quasi-coherent sheaves on a gerbe banded by a diagonalizable group decomposes according to the characters of the group. We establish the corresponding decomposition of the unbounded derived category of complexes of sheaves with quasi-coherent cohomology. This generalizes earlier work by Lieblich for gerbes over schemes whereas our gerbes may live over arbitrary algebraic stacks. \par By combining this decomposition with the semi-orthogonal decomposition for a projectivized vector bundle, we deduce a semi-orthogonal decomposition of the derived category of a family of Brauer-Severi varieties whose components can be described in terms of twisted sheaves on the base. This reproves and generalizes a result of Bernardara.