Contents Online

# Homology, Homotopy and Applications

## Volume 25 (2023)

### Number 1

### Cyclic $A_\infty$-algebras and cyclic homology

Pages: 287 – 318

DOI: https://dx.doi.org/10.4310/HHA.2023.v25.n1.a15

#### Author

#### Abstract

We provide a new description of the complex computing the Hochschild homology of an $H$-unitary $A_\infty$-algebra $A$ as a derived tensor product $A \oplus^\infty_{A^\epsilon}$ such that: **(1)** there is a canonical morphism from it to the complex computing the cyclic homology of $A$ that was introduced by Kontsevich and Soibelman, **(2)** this morphism induces the map $I$ in the well-known SBI sequence, and **(3)** $H^0 \left( (A \oplus^\infty_{A^\epsilon} A)^\# \right)$ is canonically isomorphic to the space of morphisms from $A$ to $A^\#$ in the derived category of $A_\infty$-bimodules. As direct consequences we obtain previous results of Cho and Cho–Lee, as well as the fact that Koszul duality establishes a bijection between (resp., almost exact) $d$-Calabi–Yau structures and (resp., strong) homotopy inner products, extending a result proved by Van den Bergh.

#### Keywords

dg (co)algebra, $A_\infty$-algebra, Calabi–Yau, Koszul duality

#### 2010 Mathematics Subject Classification

16E05, 16E40, 16E45, 16T15

Received 29 October 2021

Received revised 1 April 2022

Accepted 7 April 2022

Published 26 April 2023