This paper is concerned with a minimal resolution of the PROP for bialgebras (Hopf algebras without unit, counit and antipode). We prove a theorem about the form of this resolution (Theorem 15) and give, in Section 5, a lot of explicit formulas for the di

Given a commutative ring k, a group G and an element gset membership, variantG of infinite order with centralizer C(g), we study the inverse systemcdots, three dots, centeredlong right arrowH2n(C(g)/left angle bracketgright-pointing angle bracket,k)long right arrowH2n-2(C(g)/left angle bracketgright-pointing angle bracket,k)long right arrowcdots, three dots, centeredarising from Burgheleaʹs computation [D. Burghelea, The cyclic homology of group rings, Comment. Math. Helv. 60 (1985) 354–365] of the cyclic homology of the group algebra kG and Connes’ periodicity operator S:HC2n(kG)long right arrowHC2n-2(kG). A vanishing theorem for the limit of this inverse system is proved for groups in the class image introduced in Emmanouil and Passi [A contribution to Bass’ conjecture, J. Group Theory 7 (2004) 409–420], thereby contributing to a conjecture by Burghelea [The cyclic homology of group rings, Comment. Math. Helv. 60 (1985) 354–365]. The homological condition defining the class image is closely examined; in particular, it is shown that this class properly contains the class studied in Emmanouil [On a class of groups satisfying Bass’ conjecture, Invent. Math. 132 (1998) 307–330].