Generators vs. classical generators in derived categories of curves

TL;DR

This paper discusses the difference between generators and classical generators in derived categories of curves, providing an example of a non-classical generator on curves of genus at least 2 and reviewing related known results.

Contribution

It presents a specific example of a non-classical generator in the derived category of coherent sheaves on higher genus curves and reviews the current understanding of generators in this context.

Findings

Existence of non-classical generators on curves of genus ≥ 2

Classical generators are a subset of generators in derived categories

Overview of known results and open questions about generators on 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.