DG categories and exceptional collections

TL;DR

This paper explores DG categories associated with full exceptional collections on varieties, demonstrating their finite-dimensional morphism spaces, behavior under mutations, and applications to non-commutative deformations of rational surfaces.

Contribution

It shows that the DG category from a full exceptional collection has finite-dimensional morphisms and provides an algorithm to compute it, advancing understanding of non-commutative deformations.

Findings

DG categories have finite-dimensional morphism spaces

Algorithm for computing DG categories with vanishing Ext^k for k > 1

Application to non-commutative deformation of rational surfaces

Abstract

Bondal and Kapranov describe how to assign to a full exceptional collection on a variety X a DG category C such that the bounded derived category of coherent sheaves on X is equivalent to the bounded derived category of C. In this paper we show that the category C has finite dimensional spaces of morphisms. We describe how it behaves under mutations and present an algorithm allowing to calculate it for full exceptional collections with vanishing Ext^k groups for k > 1. Finally, we use it to describe an example of a non-commutative deformation of certain rational surfaces.