Abstract
A description of how to assign to a full exceptional collection on a variety X X a DG category C \mathcal {C} such that the bounded derived category of coherent sheaves on X X is equivalent to the bounded derived category of C \mathcal {C} is given in a 1990 work by Bondal and Kapranov, Framed triangulated categories. In this paper we show that the category C \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 ^k groups for k > 1 k>1 .
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