Homology, Homotopy and Applications
Volume 25 (2023)
Number 1
Cyclic -algebras and cyclic homology
Pages: 287 – 318
DOI: https://dx.doi.org/10.4310/HHA.2023.v25.n1.a15
Author
Estanislao Herscovich (Institut Fourier, Labortoire de Mathématiques, Université Grenoble Alpes, Grenoble, France)
Abstract
We provide a new description of the complex computing the Hochschild homology of an -unitary -algebra as a derived tensor product such that: (1) there is a canonical morphism from it to the complex computing the cyclic homology of that was introduced by Kontsevich and Soibelman, (2) this morphism induces the map in the well-known SBI sequence, and (3) is canonically isomorphic to the space of morphisms from to in the derived category of -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) -Calabi–Yau structures and (resp., strong) homotopy inner products, extending a result proved by Van den Bergh.
Keywords
dg (co)algebra, -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