Decompositions of derived categories of gerbes and of families of Brauer-Severi varieties
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 ge...
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Zusammenfassung: | 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.
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 familiy 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. |
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DOI: | 10.48550/arxiv.1901.08945 |