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The Hochschild homology of a noncommutative symmetric quotient stack

Rina Anno, Vladimir Baranovsky, Timothy Logvinenko

TL;DR

This work develops a noncommutative analogue of Baranovsky’s orbifold decomposition for Hochschild homology by studying symmetric powers of a small DG category ${\mathcal A}$ as noncommutative symmetric quotient stacks. It establishes an explicit isomorphism $\bigoplus_{n\ge 0} \mathrm{HH}_\bullet(\mathrm{Sym}^n {\mathcal A}) \cong S^*(\mathrm{HH}_\bullet({\mathcal A}) \otimes t \mathbb{k}[t])$, and constructs mutually inverse Hochschild-complex maps that realize the decomposition on the chain level. These foundations enable transferring rich algebraic structures: the total Hochschild homology becomes the Fock space for the Heisenberg algebra $H_{\mathrm{HH}_\bullet({\mathcal A})}$ and acquires a natural Hopf-algebra and a free $\\lambda$-ring structure. The results generalize equivariant- and orbifold-type decompositions to the noncommutative setting and connect to K-theoretic analogues, offering explicit computational tools and a framework for noncommutative orbifold cohomology theories of DG categories.

Abstract

We prove an orbifold type decomposition theorem for the Hochschild homology of the symmetric powers of a small DG category $\mathcal{A}$. In noncommutative geometry, these can be viewed as the noncommutative symmetric quotient stacks of $\mathcal{A}$. We use this decomposition to show that the total Hochschild homology of the symmetric powers of $\mathcal{A}$ is isomorphic to the symmetric algebra $S^*(\mathrm{HH}_\bullet(\mathcal{A}) \otimes t \mathbb{k}[t])$. Our methods are explicit - we construct mutually inverse homotopy equivalences of the standard Hochschild complexes involved. These explicit maps are then used to induce from the symmetric algebra onto the total Hochschild homology the structures of the Fock space for the Heisenberg algebra of $\mathcal{A}$, of a Hopf algebra, and of a free $λ$-ring generated by $\mathrm{HH}_\bullet(\mathcal{A})$.

The Hochschild homology of a noncommutative symmetric quotient stack

TL;DR

This work develops a noncommutative analogue of Baranovsky’s orbifold decomposition for Hochschild homology by studying symmetric powers of a small DG category as noncommutative symmetric quotient stacks. It establishes an explicit isomorphism , and constructs mutually inverse Hochschild-complex maps that realize the decomposition on the chain level. These foundations enable transferring rich algebraic structures: the total Hochschild homology becomes the Fock space for the Heisenberg algebra and acquires a natural Hopf-algebra and a free -ring structure. The results generalize equivariant- and orbifold-type decompositions to the noncommutative setting and connect to K-theoretic analogues, offering explicit computational tools and a framework for noncommutative orbifold cohomology theories of DG categories.

Abstract

We prove an orbifold type decomposition theorem for the Hochschild homology of the symmetric powers of a small DG category . In noncommutative geometry, these can be viewed as the noncommutative symmetric quotient stacks of . We use this decomposition to show that the total Hochschild homology of the symmetric powers of is isomorphic to the symmetric algebra . Our methods are explicit - we construct mutually inverse homotopy equivalences of the standard Hochschild complexes involved. These explicit maps are then used to induce from the symmetric algebra onto the total Hochschild homology the structures of the Fock space for the Heisenberg algebra of , of a Hopf algebra, and of a free -ring generated by .
Paper Structure (24 sections, 16 theorems, 135 equations)

This paper contains 24 sections, 16 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. DG categories, modules and tensor products
  4. Equivariant DG categories
  5. Partitions
  6. Symmetric powers of DG categories
  7. Hochschild homology of a bimodule
  8. Shuffles
  9. Künneth isomorphism
  10. Shuffle product with the constants
  11. Long cycle case
  12. Notation
  13. Map $f$
  14. Map $g$
  15. Homotopy $\Phi$
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let ${\mathcal{A}}$ be a small DG category and $n \geq 0$. The following compositions are mutually inverse isomorphisms: where $\nu$ and $\xi$ are the maps explicitly constructed in §section-from-hh-of-quotient-stack-to-twisted-hhs and $f_{\underline{\lambda}}$ and $g_{\underline{\lambda}}$ -- in §section-from-underline-n-twisted-hh-to-sym-underline-n-hh.

Theorems & Definitions (53)

  • Theorem 1.1: see Theorem \ref{['theorem-noncommutative-baranovsky-decomposition']}
  • Theorem 1.2: see Theorem \ref{['theorem-total-hh-of-sym-stacks-iso-to-sym-algebra-of-hh-otimes-tkt']}
  • Theorem 1.3: see Theorem \ref{['theorem-hopf-algebra-structure-on-the-total-hh-of-symn-a']}
  • Definition 2.1: GyengeKoppensteinerLogvinenko-TheHeisenbergCategoryOfACategory, Defn. 4.45
  • Lemma 2.2: GyengeKoppensteinerLogvinenko-TheHeisenbergCategoryOfACategory, Lemma 4.41
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • ...and 43 more