Generators and representability of functors in commutative and noncommutative geometry

TL;DR

This paper establishes conditions under which certain triangulated categories are saturated, showing that derived categories of coherent sheaves on smooth proper varieties are saturated, unlike some analytic surface categories.

Contribution

It proves that bounded derived categories of coherent sheaves on smooth proper varieties have strong generators, ensuring their saturation in both commutative and noncommutative geometry.

Findings

Bounded derived categories of coherent sheaves on smooth proper varieties are saturated.

Existence of a strong generator implies saturation of the category.

Non-saturation occurs in certain analytic surface categories without curves.

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 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, hence saturated. In contrast the similar category for a smooth compact analytic surface with no curves is not saturated.