Reflexive and dihedral (co)homology of a pre-additive category
The goal of this paper is to offer the calculation of the dihedral homology, \(HD_*\), of the free product of algebras associated with a pre-additive \(k\)-category, where \(k\) has characteristic zero. The paper opens with a recollection of the definition of dihedral homology [\textit{J.-L. Loday}, Adv. Math. 66, 119-148 (1987; Zbl 0627.18006), \textit{R. L. Krasauskas, S. V. Lapin} and \textit{Yu. P. Solov'ev}, Math. USSR, Sb. 61, No. 1, 23-47 (1988); translation from Mat. Sb., Nov. Ser. 133(175), No. 1, 25-48 (1987; Zbl 0628.18008); \textit{G. M. Lodder}, Proc. Lond. Math. Soc., III. Ser. 60, No. 1, 201-224 (1990; Zbl 0691.55006)]. Then is recalled that the cyclic homology of an algebra \(A\) with involution (over a characteristic zero field) splits as the direct sum of \(HD_*(A)\) and \(^-HD_*(A)\), where \(^-HD_*(A)\) is the dihedral homology of \(A\) with the opposite sign for the involution. For \(A\), \(B\), \(C\) involutive algebras associated with a pre-additive category as above, \(A*B\) denotes the free product of \(A\) and \(B\) over \(C\), which naturally inherits an involution. If \({\text{Tor}}^C_i (A,A)\), \({\text{Tor}}^C_i(A,B)\), \({\text{Tor}}^C_i(B, B)\) are all zero for \(i > 0\), then \(HD_*( A*B) \oplus HD_*(C)\) is expressed as the direct sum \(HD_*(A) \oplus HD_*(B) \oplus Z\), where \(Z\) denotes the hyper-homology of the dihedral groups \(D_{n+1}\), \(n \geq 0\), with coefficients in a chain complex involving the free involutive resolutions of the algebras \(A\) and \(B\).
scientific article; zbMATH DE number 1740564