A decomposition theorem for the Hochschild homology of symmetric powers of a dg category
Ville Nordstrom
TL;DR
The paper resolves the Belmans–Fu–Krug conjecture by proving that the Hochschild homology of symmetric powers of a small dg category decomposes as a symmetric algebra on the direct sum of Hochschild homology of the original category. The authors decompose $HH_\bullet(\text{Sym}^n(\mathscr{C}))$ into contributions from conjugacy classes in $S_n$, apply a Künneth-type factorization, and crucially show $HH_\bullet(\mathscr{C}^{\otimes i}, \sigma_i) \cong HH_\bullet(\mathscr{C})$, which reduces the computation to $HH_\bullet(\mathscr{C})$ repeatedly. The main result, $\oplus_{n\ge0} HH_\bullet(\text{Sym}^n(\mathscr{C})) t^n \cong \text{Sym}^\bullet\big(\bigoplus_{i\ge1} HH_\bullet(\mathscr{C}) t^i\big)$, is shown to be natural in $\mathscr{C}$ and aligns with recent related work. This decomposition yields explicit, computable descriptions of Hochschild invariants for symmetric powers and connects to broader algebraic structures, such as Hopf-algebra-like behaviors, in the derived setting.
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 $\mathscr{C}$.