Excision in Cyclic Homology Theories

this paper we describe a new approach to excision in periodic cyclic homology. It applies also to cyclic homology theories for topological algebras and establishes excision in entire, asymptotic and local bivariant cyclic cohomology. Among other things we obtain sharp upper bounds for the dimension shift of periodic cyclic cohomology under the boundary map associated to an extension of algebras. Ever since the invention of cyclic homology by Connes and independently by Tsygan the problem of excision played a central role in the theory. It is concerned with the question of to what extent an extension 0 ! I ! A ! B ! 0 of algebras induces natural long exact sequences on cyclic homology groups. To put our approach into perspective we recall the previous work on excision in cyclic homology. The first results on this problem were obtained by Wodzicki [Wo]. In 1989 he introduced the notion of H-unitality (homological unitality) for algebras and showed that excision in cyclic homology holds for all (linearly split) extensions, 0 ! I ! A ! B ! 0 with fixed kernel I, if and only if I is H-unital as an algebra. This condition is however rarely satisfied in practice. For example excision in cyclic homology does not hold for nilpotent extensions because nilpotent algebras are neve