This monograph is concerned with the notions of ditalgebras (an acronym for “differential tensor algebras”) and the study of their categories of modules. It involves reduction techniques which have proved to be very useful in the development of the theory of representation of finite-dimensional algebras. Our aim has been to present in a systematic, elementary and self-contained as possible way some of the main results obtained with these methods. They were originally introduced by the Kiev School in representation theory of algebras, in an attempt to formalize and generalize matrix problems methods.

The presentation given here has many common features with the original one of A. V. Roiter and M. Kleiner [46], in terms of differential graded categories, as well as with the formulation given by Y. Drozd [28] (and further developed by W. Crawley-Boevey [19] and [20]), in terms of bocses. It is clear that some applications of these techniques, notably in the study of coverings in representation theory of algebras, will require the categorical formulation of the theory, as suggested in [30]. However, for the sake of simplicity, we preferred to work here in the more concrete ring theoretical context of ditalgebras. We assume from the reader some familiarity with the basics of representation theory of algebras and homological algebra (including the basics of the theory of additive categories with exact structures), which can be obtained from the first sections of [29], [47] and [3] (respectively, [32] and [27]).