For a given endomorphism on a unitary κ-algebra, Α, with κ in the center of Α, 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.
Shapiro, Jack M., "Relations Between Twisted Derivations and Twisted Cyclic Homology" (2012). Mathematics Faculty Publications. 4.