Symmetric products of dg categories and semi-orthogonal decompositions

Abstract

In this article, we investigate semi-orthogonal decompositions of the symmetric products of dg-enhanced triangulated categories. Given a semi-orthogonal decomposition $\mathcal{D}=⟨\mathcal{A},\mathcal{B}⟩$, 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.

arXiv comment14 pages, improvements based on referees' comments, Section 4 in the previous version is removed due to an error in the proof of Proposition 4.1