Homology, Homotopy and Applications

Volume 25 (2023)

Number 1

Cyclic A-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 H-unitary A-algebra A as a derived tensor product AAϵ 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) H0((AAϵA)#) is canonically isomorphic to the space of morphisms from A to A# in the derived category of A-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-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