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.

Homology of systemic modules

“…We can define a specific chain complex from a given module M with negation, applying symmetrization to the classical theory given in [9,35,48]. We define (−) 1 Tensor products over semirings parallel tensor products over rings, and are well studied in the literature [36,37,38]; the systemic version is given in [45, § 6.4], where it is stipulated that (−)(b ⊗ b ′ ) = ((−)b) ⊗ b ′ for all b, b ′ .…”

“…Example 5.9. Here is the systemic version of [9], [48]. Given a semiring M which is a module with negation over a commutative semiring system (A, T , (−), ), define The version d n : M ⊗n → M ⊗n−1 under symmetrization for an arbitrary semi-algebra over a semiring is to symmetrize the differential map, i.e., put the positive parts in the first component and the negative parts in the second component, taking…”

Cyclic vs mixed homology

“…To generalise the theory of Cuntz and Quillen (which concerns the case where σ = id) to this setting was one of our original aims, motivated in particular by Shapiro's extension [16] of Karoubi's noncommutative De Rham theory.…”

“…Early versions and special instances of the main results of our paper were key tools in the computation of the twisted cyclic homology of quantum SL(2) due to Hadfield and the first author [9]. Another source of motivation was the work of Shapiro [16] who investigated the approach via noncommutative differential forms. Although the results as formulated here are fairly technical, we felt it worthwhile to present them in full generality from the viewpoint of mixed complexes and hope they will find new applications in other settings in the future.…”

Twisted Cyclic Homology And Crossed Product Algebras

“…It will then follow that if order(g) = r, with r invertible in k, then HC g * (A) ≃ HC g n * (A), for g raised to any power n, where (n, r) = 1. For the case where G is a finite group and Q ⊂ k we can generalize the proceedure used in [4] for the twisted de Rham homology of A, to define HDR G * (A), the G-de Rham homology of A, and from that get a Karoubi exact sequence…”