In this work, we give a characterization of free crossed modules and also get a relation between projective crossed modules and the cyclic homology of associative algebras by using Hopf-type formulas. | Turk J Math (2016) 40: 995 – 1003 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Projective crossed modules of algebras and cyclic homology Elif ILGAZ∗ Department of Mathematics and Computer Science, Osmangazi University, Eski¸sehir, Turkey Received: • Accepted/Published Online: • Final Version: Abstract: In this work, we give a characterization of free crossed modules and also get a relation between projective crossed modules and the cyclic homology of associative algebras by using Hopf-type formulas. Key words: Crossed module of algebras, free crossed module, projective crossed module, cyclic homology 1. Introduction Crossed modules were first defined by Whitehead [19]. Areas in which crossed modules have been applied include the theory of group presentations, algebraic K-theory, and homological algebra. The crossed module theory has been deeply analyzed by Brown et al.’s book “Nonabelian Algebraic Topology”, [5], and in the book by Porter “The Crossed Menagerie”[16]. As an application of cyclic homology of associative algebras, the Hopf-type formulas for the cyclic homology given in Brown and Ellis [4] is developed by using the way of n-fold ˇ Cech derived functors in [7]. In this paper, we consider the work by Ratcliffe [17], involving a homological characterization of free and projective crossed modules of groups. He mainly proved that C is a projective crossed G -module if and only if C/C 2 is projective as a coker∂ -module and the second homology morphism ∂∗ : H2 C −→ H2 N is trivial where N = ∂(C). In this paper, we want to describe an analogous philosophy for crossed modules of associative algebras, which will be called crossed modules of algebras hereafter. The results are important for examining the first cyclic homology of associative algebras, in terms of Hopf-type formulas. It is well known that all free modules are projective .
