Excision in cyclic homology theories

Abstract

In this paper we establish excision for Z/2Z-graded cyclic homology theories, including periodic, entire, asymptotic and local bivariant cyclic cohomology. Our work is a modification of the approach of Cuntz-Quillen [CQ] to excision in bivariant periodic cyclic homology and makes also use of the work of Wodzicki [Wo]. We study the dimension shift of periodic cyclic cohomology under the boundary map associated to an extension of algebras and obtain the possible values of this dimension shift. Excision in entire cyclic cohomology is used to calculate these groups in a number of hitherto unknown cases. Finally a bivariant and multiplicative Chern-Connes character on Kasparov’s bivariant K-theory [Ka] with values in bivariant local cyclic cohomology is constructed. To put our approach into perspective we recall the previous work on excision in cyclic homology. 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