Generators vs. classical generators in derived categories of curves
Dmitrii Pirozhkov
TL;DR
The paper investigates the distinction between generators and classical generators in the derived category $D^b_{\mathrm{coh}}(C)$ of a smooth projective curve $C$, proving a striking criterion: an object $E$ is a generator if and only if it is not semistable, with the easy direction following from the Riemann–Roch formula and semi-orthogonality considerations. Building on this, it provides an explicit, fully worked example on curves of genus $g\ge 2$: a line bundle $L$ of degree $1$ with $h^0(L)=0$ yields $\mathcal{O}_C \oplus L$ which is a generator but not a classical generator, showcasing a non-classical generator in $D^b_{\mathrm{coh}}(C)$. The note also surveys known results on generators and classical generators on curves, discusses how the phenomena behave in families, and presents open questions, such as when slope gaps guarantee classical generators and how generating time can be bounded. The results emphasize that the two notions diverge even in well-behaved geometric settings and have implications for moduli problems and the structure of 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.