Double derivations and Cyclic homology

Details Double derivations and Cyclic homology Double derivations and Cyclic homology

2005-05-12

arxiv.org/abs/math/0505236v3

Mathematics

K-Theory and Homology

Entirely rewritten, additional results added, exposition simplified; 31pp

Scientific paper

We give a new construction of cyclic homology of an associative algebra A that does not involve Connes' differential. Our approach is based on an extended version of the complex \Omega A, of noncommutative differential forms on A, and is similar in spirit to the de Rham approach to equivariant cohomology. Indeed, our extended complex maps naturally to the equivariant de Rham complex of any representation scheme Rep A. We define cyclic homology as the cohomology of the total complex (\Omega A)[t], d+t \cdot i, arising from two anti-commuting differentials, d and i, on \Omega A. The differential d, that replaces the Connes differential B, is the Karoubi-de Rham differential. The differential i that replaces the Hochschild differential b, is a map analogous to contraction with a vector field. This new map has no commutative counterpart.

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