Rina Anno - The Hochschild homology of a noncommutative symmetric quotient stack скачать в хорошем качестве

Rina Anno - The Hochschild homology of a noncommutative symmetric quotient stack

Rina Anno (Kansas State University); February 26, 2026 For a small DG category $A$, its symmetric power $S^nA$ may be considered a noncommutative symmetric quotient stack of $A$. We establish an isomorphism between $\bigoplus HH_\bullet(S^nA)$ and the symmetric algebra $S^*(HH_\bullet(A) \otimes t k[t])$ by chaining explicit maps of complexes. These graded vector spaces being isomorphic has been established before by Baranovsky in the commutative case and conjectured by Belmans, Fu, and Krug in the form that we prove it. The explicit nature of the isomorphism allows us to transfer a number of structures from the symmetric algebra to $\bigoplus HH_\bullet(S^nA)$, since the former is a Hopf algebra, the Fock space for the Heisenberg algebra of $A$, and a $\lambda$-ring. In this talk, I will go over (some of the) history of the results this one is building upon, describe the quasiisomorphism, and compare the result to the (much more studied) commutative case. This talk is based on a joint work with V. Baranovsky and T. Logvinenko, https://arxiv.org/abs/2512.25039