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This volume provides a systematic presentation of the theory of differential tensor algebras and their categories of modules. It involves reduction techniques which have proved to be very useful in the development of representation theory of finite dimensional algebras. The main results obtained with these methods are presented in an elementary and self contained way. The authors provide a fresh point of view of well known facts on tame and wild differential tensor algebras, on tame and wild algebras, and on their modules. But there are also some new results and some new proofs. Their approach presents a formal alternative to the use of bocses (bimodules over categories with coalgebra structure) with underlying additive categories and pull-back reduction constructions. Professional mathematicians working in representation theory and related fields, and graduate students interested in homological algebra will find much of interest in this book.

Abstract. This paper deals with the representation theory of strongly locally finite quivers. We first study some properties of the finitely presented or copresented representations, and then construct in the category of locally finite dimensional representations some almost split sequences which start with a finitely co-presented representation and end with a finitely presented representation. Furthermore, we obtain a general description of the shapes of the Auslander-Reiten components of the category of finitely presented representations and prove that the number of regular Auslander-Reiten components is infinite if and only if the quiver is not of finite or infinite Dynkin type. In the infinite Dynkin case, we shall give a complete list of the indecomposable representations and an explicit description of the Auslander-Reiten components. Finally, we apply these results to study the Auslander-Reiten theory in the derived category of bounded complexes of finitely presented representations.

Abstract. We extend the Galois covering theory introduced by BongartzGabriel for skeletal linear categories to general linear categories. We show that a Galois covering between Krull-Schmidt categories preserves irreducible morphisms and almost splits sequences. Specializing to derived categories, we study when a Galois covering between locally bounded linear categories induces a Galois covering between the bounded derived categories of finite dimensional modules. As an application, we show that each locally bounded linear category with radical squared zero admits a gradable Galois covering, which induces a Galois covering between the bounded derived categories of finite dimensional modules, and a Galois covering between the Auslander-Reiten quivers of these bounded derived categories. In a future paper, this will enable us to obtain a complete description of the bounded derived category of finite dimensional modules over a finite dimensional algebra with radical squared zero.

Let A be an additive k-category, k a commutative artinian ring and n > 1. We denote by C n (A) the category of complexes X = (X i , d i X ) i∈Z in A with X i = 0 if i / ∈ {1, . . . , n}. We see that C n (A) is endowed with a natural exact structure and its global dimension is at most n − 1. In case A is a dualizing category, we prove that C n (A) has almost split sequences in the sense of [P. Dräxler, I. Reiten, S.O. Smalø, Ø. Solberg, Exact categories and vector space categories, with an appendix by B. Keller, Trans. Amer. Math. Soc. 351 (2) (1999) 647-682] or [R. Bautista, The category of morphisms between projective modules, Comm. Algebra 32 (11) (2004) 4303-4331]. If A is the category of finitely generated projective Λ-modules (Λ an Artin algebra), we prove that the ends of an almost split sequence are related by an Auslander-Reiten translation functor which is defined in the most general category C n (Proj Λ).

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Abstract. Let Λ be an elementary locally bounded linear category over a field with radical squared zero. We shall show that the bounded derived category D b (Mod b Λ) of finitely supported left Λ-modules admits a Galois covering which is the bounded derived category of almost finitely co-presented representations of a gradable quiver. Restricting to the bounded derived category D b (mod b Λ) of finite dimensional left Λ-modules, we shall be able to describe its indecomposable objects, obtain a complete description of the shapes of its Auslander-Reiten components, and classify those Λ such that D b (mod b Λ) has only finitely many Auslander-Reiten components.