Homotopy invariance of the cyclic homology of \(A_{\infty}\)-algebras under homotopy equivalences of \(A_{\infty}\)-algebras (Q2059045)

In [\textit{S. V. Lapin}, Math. Notes 102, No. 6, 806--823 (2017; Zbl 1381.16009); translation from Mat. Zametki 102, No. 6, 874--895 (2017)], Lapin constructed the cyclic bicomplex of an \(A_{\infty}\)-algebra over a commutative unital ring. This generalized the famous cyclic bicomplex for an associative algebra due to \textit{B. L. Tsygan} [Russ. Math. Surv. 38, No. 2, 198--199 (1983; Zbl 0526.17006)] and \textit{J.-L. Loday} and \textit{D. Quillen} [Comment. Math. Helv. 59, 565--591 (1984; Zbl 0565.17006)]. The cyclic homology of an \(A_{\infty}\)-algebra is defined as the homology of the chain complex associated to Lapin's bicomplex. In the paper under review Lapin proves that the cyclic homology of \(A_{\infty}\)-algebras is homotopy invariant under homotopy equivalences of \(A_{\infty}\)-algebras. The paper is organized as follows. Section 2 recalls the definition of a cyclic module with \(\infty\)-simplicial faces, or \(CF_{\infty}\)-module, as introduced in [\textit{S. V. Lapin}, loc. cit.]. In particular, as recalled in Section 4 of the paper under review, any \(A_{\infty}\)-algebra gives rise to a \(CF_{\infty}\)-module. The notion of homotopy equivalence of \(CF_{\infty}\)-modules is introduced. Section 3 recalls the definition of cyclic homology of \(CF_{\infty}\)-modules. It is shown that there is a cyclic homology functor from the category of \(CF_{\infty}\)-modules to the category of graded modules over the ground ring. Furthermore, it is shown that this functor sends homotopy equivalences of \(CF_{\infty}\)-modules to isomorphisms of graded modules. In Section 4, the author recalls the definition of \(A_{\infty}\)-algebra and how to construct a \(CF_{\infty}\)-module from an \(A_{\infty}\)-algebra. The results of Section 3 are then applied to give a cyclic homology functor from the category of \(A_{\infty}\)-algebras to the category of graded modules, sending homotopy equivalences of \(A_{\infty}\)-algebras to isomorphisms of graded modules.