Cyclic homology of cyclic \(\infty\)-simplicial modules
In the paper under review, the author defines cyclic $\infty$-modules, and states properties of their cyclic homology, $HC_*$, suitably defined to apply to this more general setting. Recall that a differential simplicial module $X = \{ X_{n, m} \}$, $n$, $m \in \mathbb{Z}$, $n \geq 0$, supports a simplicial structure in the subscript $n$ and a differential $d : X_{n, m} \to X_{n, m -1}$ in the subscript $m$. In previous work [\textit{S. V. Lapin}, Math. Notes 96, No. 6, 698--715 (2014; Zbl 1332.55010); translation from Mat. Zametki 96, No. 5, 709--731 (2014)], the author has generalized such modules to differential $\infty$-simplicial modules, also denoted $X$. Let $(S,\pi)$ be the base colored algebra of simplicial faces and degeneracies in the above reference. Let $S^!$ be the Koszul dual to the quadratic-scalar colored algebra $(S,\pi)$. A differential $\infty$-simplicial module $X$ is a differential module $(X, d)$ with a map \[ \psi : (S^! \otimes X)_{*,*} \to X_{*, *-1} \] satisfying two special conditions stated in the paper. Now let $C_{n+1}$ denote the cyclic group of order $n+1$ and let $k$ be a commutative ring with unit. Consider the colored algebra $k[C]$ defined by the group ring \[ k[C](n, n)_0 = k[C_{n+1}], \] and $K[C](m,n)_k = 0$ otherwise. Define a colored module $(S^!)^C$ as a certain quotient \[ k[C] \otimes S^! \otimes k[C]/\sim . \] Then a cyclic $\infty$-simplicial module, abbreviated $X$, is a differential left $k[C]$-module together with a map \[ \varphi : ( (S^!)^C \otimes_{k[C]} X )_{*,*} \to X_{*, *-1} \] satisfying two additional conditions. The author proposes a definition for $HC_*(X)$ using a $b$, $b'$ double comples in which the $n$th row contains the periodic resolution of $C_{n+1}$ applied to $\bar{X}_n = \sum_{k=0}^n X_{k, n-k}$. It might be of interest to state the definition for the cyclic homology of a cyclic $\infty$-simplicial module in terms of a derived functor. Let $\mathrm{Coinv}(\bar{X})_n$ denote the coinvariants of $\bar{X}_n$ formed by $C_{n+1}$. Proven is that over a field of characteristic zero, $HC_*(X)$ is isomorphic to the chain complex $\mathrm{Coinv}(\bar{X})$, $b$, which generalizes a result from [\textit{A. Connes}, C. R. Acad. Sci., Paris, Sér. I 296, 953--958 (1983; Zbl 0534.18009)]. Also proven is a generalization of the Connes-Tsygan long exact sequence for any cyclic $\infty$-simplicial module $X$: \[ \longrightarrow H_n(X) \longrightarrow HC_n(X) \longrightarrow HC_{n-2}(X) \longrightarrow H_{n-1}(X) \longrightarrow \ldots . \]
scientific article; zbMATH DE number 7000275
scientific article