Geodesic
Decompositions of derived categories of gerbes and of families of Brauer-Severi varieties
Documenta mathematica, Tome 26 (2021), pp. 1465-1500.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

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.
Classification : 14F08, 14A20
Keywords: semi-orthogonal decomposition, derived category, gerbe, Brauer-Severi variety, algebraic stack 
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     title = {Decompositions of derived categories of gerbes and of families of {Brauer-Severi} varieties},
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Bergh, Daniel; Schnürer, Olaf M. Decompositions of derived categories of gerbes and of families of Brauer-Severi varieties. Documenta mathematica, Tome 26 (2021), pp. 1465-1500. http://geodesic.mathdoc.fr/item/DOCMA_2021__26__a18/