DG categories and exceptional collections

Bodzenta, Agnieszka

doi:10.1090/S0002-9939-2014-12420-8

DG categories and exceptional collections
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Proc. Amer. Math. Soc. 143 (2015), 1909-1923 Request permission

Abstract:

A description of how to assign to a full exceptional collection on a variety $X$ a DG category $\mathcal {C}$ such that the bounded derived category of coherent sheaves on $X$ is equivalent to the bounded derived category of $\mathcal {C}$ is given in a 1990 work by Bondal and Kapranov, Framed triangulated categories. In this paper we show that the category $\mathcal {C}$ can be chosen to have finite-dimensional spaces of morphisms. We describe how it behaves under mutations and present an algorithm allowing us to calculate it for full exceptional collections with vanishing Ext$^k$ groups for $k>1$.

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