A decomposition theorem for the Hochschild homology of symmetric powers of a dg category

TL;DR

This paper proves a conjecture that the Hochschild homology of symmetric powers of a dg category decomposes into parts determined solely by the original category's Hochschild homology, revealing a structural simplification.

Contribution

It establishes a decomposition theorem for Hochschild homology of symmetric powers of dg categories, confirming a conjecture and clarifying the homological structure.

Findings

Hochschild homology of symmetric powers decomposes into simpler components

The decomposition depends only on the original dg category's Hochschild homology

The result confirms a conjecture by Belmans, Fu, and Krug

Abstract

We prove a conjecture by Belmans, Fu and Krug concerning the Hochschild homology of the symmetric powers of a small dg category $\mathscr{C}$. More precisely, we show that these groups decompose into pieces that only depend on the Hochschild homology of the dg category $\mathscr{C}$.