Homological algebra of \(N\)-complexes and Hochschild homology of roots of unity (Q5931089)

This article develops systematically the foundations of homological algebra for \(N\)-complexes. Recall that \(N\)-complexes are graded modules provided with a degree \(-1\) endomorphism \(d\) such that \(d^N=0\). These objects have been also studied in the work of Dubois-Violette et al. and Kapranov on the quantum differential calculus [see, e.g., \textit{M. Dubois-Violette} and \textit{R. Kerner}, ``Universal \(Z_N\)-graded differential calculus'', J. Geom. Phys. 23, No. 3-4, 235-246 (1997; Zbl 1004.46045); \textit{M. Dubois-Violette}, ``Generalized differential spaces with \(d^N=0\) and the \(q\)-differential calculus'', Czech. J. Phys. 46, No. 12, 1227-1233 (1996; Zbl 0953.18004)]. The authors introduce for example the notions of projective or injective \(N\)-resolutions, as well as \(p\text{Tor}\) and \(p\text{Ext}\) groups that share some properties with the usual Tor and Ext groups.NEWLINENEWLINENEWLINEThe first motivation of this work is provided by the Hochschild complex of an associative algebra \(A\). When the powers of \(-1\) in the definition of the Hochschild boundary \(b\) are replaced by the powers of a scalar \(q\), the \(q\)-version \(b_q\) of the boundary is such that \(b_q^N=0\) in the following cases: \(q\) is a primitive \(N\)-root of unity or \(q=1\), \(N\) is prime and \(N=0\) in \(A\). The main theorem on Hochschild homology of the article relates the \(q\)-version to the usual one as follows. Under the above assumptions on \(N\) and \(q\): for each \(n\geq 0\) and each \(p=1,\dots,N-1\): NEWLINE\[NEWLINEpHH_n(A)\cong HH_{2(n-p+1)/N}(A) \;\text{if} n+1\equiv p \bmod N,NEWLINE\]NEWLINE NEWLINE\[NEWLINEpHH_n(A)\cong HH_{(2n+2-N)/N}(A) \;\text{if} n+1\equiv 0\bmod N,NEWLINE\]NEWLINE and \(pHH_n(A)=0\) else.