process

176 C. Voigtalgebra A. The space Cn H .A/ in degree n of this cyclic module is the coinvariant spaceof H ˝ A˝nC1 with respect to a certain action of H . Recall that the coinvariant spaceV H associated to an H -module V is the quotient of V by the linear subspace generatedby all elements of the form t v .t/v. Using the natural identification n H .A/ D H ˝ A˝nC1 ˚ H ˝ A˝nthe action considered by Akbarpour and Khalkhali corresponds precisely to the actionof H on the first summand in this decomposition. Hence Cn H .A/ can be identifiedas a direct summand in the coinvariant space H n .A/ H . Moreover, the cyclic modulestructure of C H .A/ reproduces the boundary operators b and B of H .A/ H . We pointout that the relation T D id holds on the coinvariant space H .A/ H which means thatthe latter is always a mixed complex.It follows that there is a natural isomorphism of the cyclic type homologies associatedto the cyclic module C H .A/ and the mixed complex H .A/ H , respectively. Notealso that the complementary summand of C H .A/ in H .A/ H is obtained from thebar complex of A tensored with H . Since A is assumed to be a unital H -algebra, thiscomplementary summand is contractible with respect to the differential induced fromthe Hochschild boundary of H .A/ H .In the cohomological setting Akbarpour and Khalkhali introduce a cocyclic moduleCH .A/ for every unital H -module algebra. The definition of this cocyclic modulegiven in [2] is not literally dual to the one of the cyclic module C H .A/. In order toestablish the connection to our constructions let first A be an arbitrary H -algebra. Wedefine a modified action of H on H .A/ by the formulat ı .x ˝ !/ D t .2/ xS 1 .t .3/ / ˝ t .1/ !and write H .A/ for the space H .A/ equipped with this action. Let us comparethe modified action with the original actiont .x ˝ !/ D t .3/ xS.t .1/ / ˝ t .2/ !introduced in Section 6. In the space H .A/ H of coinvariants with respect to theoriginal action we havet ı .x ˝ !/ D t .2/ xS 1 .t .3/ / ˝ t .1/ ! D t .4/ xS 1 .t .5/ /t .1/ S.t .2/ / ˝ t .3/ !D t .2/ .xS 1 .t .3/ /t .1/ ˝ !/ D xS 1 .t .2/ /t .1/ ˝ ! D .t/ x ˝ !which implies that the canonical projection H .A/ ! H .A/ H factorizes over thecoinvariant space H .A/ Hwith respect to the modified action. Similarly, in the coinvariantspace H .A/ Hwe havet .x ˝ !/ D t .3/ xS.t .1/ / ˝ t .2/ ! D t .3/ xS.t .1/ /t .5/ S 1 .t .4/ / ˝ t .2/ !D t .2/ ı .xS.t .1/ /t .3/ ˝ !/ D xS.t .1/ /t .2/ ˝ ! D .t/ x ˝ !:As a consequence we see that the identity map on H .A/ induces an isomorphism H .A/ H Š H .A/ H