Reflexive and dihedral (co)homology of a pre-additive category. (English) Zbl 1002.18014

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 [J.-L. Loday, Adv. Math. 66, 119-148 (1987; Zbl 0627.18006), R. L. Krasauskas, S. V. Lapin and 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); 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 \(B\).


18G60 Other (co)homology theories (MSC2010)
16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
55P91 Equivariant homotopy theory in algebraic topology
55Q91 Equivariant homotopy groups
18E05 Preadditive, additive categories
55N91 Equivariant homology and cohomology in algebraic topology
Full Text: DOI EuDML