Symmetric products of dg categories and semi-orthogonal decompositions

TL;DR

This paper studies how semi-orthogonal decompositions of dg categories behave under symmetric products, providing new constructions and applications to derived categories of Hilbert schemes of points on surfaces.

Contribution

It constructs semi-orthogonal decompositions of symmetric products of dg categories based on existing decompositions, answering a question by Ganter--Kapranov and applying to Hilbert schemes.

Findings

Constructed semi-orthogonal decompositions for symmetric products of dg categories.

Connected the results with the derived McKay correspondence.

Derived new decompositions for categories of Hilbert schemes of points.

Abstract

In this article, we investigate semi-orthogonal decompositions of the symmetric products of dg-enhanced triangulated categories. Given a semi-orthogonal decomposition $\mathcal{D}=\langle \mathcal{A}, \mathcal{B} \rangle$, we construct semi-orthogonal decompositions of the symmetric products of $\mathcal{D}$ in terms of that of $\mathcal{A}$ and $\mathcal{B}$. This was originally stated by Galkin--Shinder, and answers the question raised by Ganter--Kapranov. Combining the above result with the derived McKay correspondence, we obtain various interesting semi-orthogonal decompositions of the derived categories of the Hilbert schemes of points on surfaces.