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**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 \(B\).

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 \(B\).

Reviewer: Jerry M.Lodder (Las Cruces)

### MSC:

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 |