Excision in Hopf Cyclic Homology

Abstract

Abstract. In this paper, we show that, under natural homological conditions, Hopf cyclichomology theory has excision. Mathematics Subject Classifications (2000): 16W30, 19D55, 57T05, 58B34. Key words: cyclic homology, Hopf algebras. 1. Introduction Cyclic cohomology theory for Hopf algebras was introduced by Connesand Moscovici in their ground breaking paper on transverse index theory[4] (cf. also [3, 5, 6] for a recent survey and new results). This theory can beregarded as the right noncommutative analogue of Lie algebra homologybecause of existence of a characteristic map for actions of Hopf algebrasand since it reduces to Lie algebra homology for Hopf algebras associ-ated to Lie algebras. This should be compared with the role played byordinary cyclic cohomology of noncommutative algebras and its reductionto de Rham homology for smooth commutative algebras [2]. The alge-braic underpinnings of Hopf cyclic theory however turned out to be morecomplicated. It is now understood as a special case of a general theorydeveloped in the series of papers [11, 12, 15, 16] (with the help of [1])for (co)algebras endowed with a (co)action of a Hopf algebra and where,among other things, the right notion of coefficients was introduced (cf. also[14] for a recent survey). This theory was later extended by the first author[13] to bialgebras and to a much larger class of coefficients. This extensionis rather surprising since existence of a bijective antipode was essential inconstructing the cyclic and cocyclic complexes developed for Hopf cyclichomology of both variants. By bialgebra or Hopf cyclic homology we nowmean the theory expounded in these papers mentioned above.The main results of the present paper are two excision theorems(Theorems 4.13 and 7.7) for Hopf cyclic homology of module coal-gebras and comodule algebras, respectively. A basic property of cyclic(co)homology for algebras, and coalgebras is excision. If by cyclic theory