Generators vs. classical generators in derived categories of curves

Abstract

This is mostly an expository note about an example communicated to the author by Aise Johan de Jong. In a triangulated category $T$ an object $G$ is said to be a classical generator when the smallest triangulated subcategory containing $G$ coincides with the whole $T$, and it is said to be a generator when the orthogonal complement to $G$ in $T$ is zero, i.e., when any non-zero object of $T$ admits a non-zero map from a shift of $G$. Any classical generator is a generator, but not vice versa. We discuss a simple algebro-geometric example of a non-classical generator in the derived category of coherent sheaves on any smooth proper curve of genus $g \geq 2$. We also overview what is known and what is not known, in general, about generators and classical generators on curves.