Generators and representability of functors in commutative and noncommutative geometry

Abstract

We give a sufficient condition for an Ext-finite triangulated category to be saturated. Saturatedness means that every contravariant cohomological functor of finite type to vector spaces is representable. The condition consists in the existence of a strong generator. We prove that the bounded derived categories of coherent sheaves on smooth proper commutative and noncommutative varieties have strong generators, and are hence saturated. In contrast, the similar category for a smooth compact analytic surface with no curves is not saturated.