Relations between twisted derivations and twisted cyclic homology

Abstract

For a given endomorphism on a unitary k-algebra, A, with k in the center of A, there are definitions of twisted cyclic and Hochschild homology. This paper will show that the method used to define them can be used to define twisted de Rham homology. The main result is that twisted de Rham homology can be thought of as the kernel of the Connes map from twisted cyclic homology to twisted Hochschild homology. For a noncommutative algebra A, Connes [1] and Karoubi [4] define the module of n-forms by taking the iterated tensor product of the bimodule of 1-forms. Karoubi in [4, Chapter 2] defines noncommutative de Rham homology. For the case where A is an associative unitary algebra over a commutative ring, k, containing Q, he shows that the reduced noncommutative de Rham homology is isomorphic to the kernel of the Connes map B from HCn(A), reduced cyclic homology, to HHn+1(A), reduced Hochschild homology. The bimodule of 1-forms used comes from the derivation d : A → A⊗kA, with d(a) = 1⊗a−a⊗1. In this paper we will generalize that result to the case of twisted cyclic homology based on a given k-algebra endomorphism, h : A → A. Here the bimodule of 1-forms will come from the twisted derivation, d : A → A ⊗k A, with d(a) = 1 ⊗ a − h(a)⊗ 1. If h = id, then we have d = d. A twisted derivation is any k-linear map, d, from A to an A-bimodule such that d(ab) = d(a) · b + h(a) · d(b). The definitions for twisted Hochschild homology and twisted cyclic homology are given in [3]. As pointed out there, in the twisted case C∗(A) itself is a paracyclic object, and we need to take an appropiate quotient of C∗(A) to obtain a cyclic object. It is quite interesting that even though the intrinsic definitions of the h-twisted theories differ considerably from the classical nontwisted theories, still there is an appropriate extension of de Rahm homology to the twisted case and an extension of the maps between all the twisted theories that give us a generalization of [4, Theorem 2.15]. Theorem. Suppose A is a unitary k-algebra with Q ⊂ k. If h is a k-algebra endomorphism then the reduced twisted de Rham homology and the reduced twisted cyclic homology are related by the exact sequence 0 → HDR n(A) → HC n (A) → HH n+1(A). HH ∗ (A) is the reduced twisted Hochschild homology and HC h,λ ∗ (A) is the reduced twisted cyclic homology based on the Connes complex {C n (A), b}. Details of the Connes complex together with the definition of HDR n(A) will be given Received by the editors March 11, 2009 and, in revised form, March 15, 2011. 2010 Mathematics Subject Classification. Primary 16E40; Secondary 16T20. c ©2011 American Mathematical Society Reverts to public domain 28 years from publication