The spectrum of a module category (Q2704687)

This monograph is a study of the Ziegler spectrum with particular reference to finite-dimensional algebras and is very much based on functor-category-theoretic methods.NEWLINENEWLINENEWLINEThe Ziegler spectrum of a ring \(R\) is a topological space which has, for its points, the isomorphism classes of indecomposable algebraically compact (= pure-injective) (right) \(R\)-modules. It arise in Ziegler's study [\textit{M. Ziegler}, Ann. Pure Appl. Logic 26, 149-213 (1984; Zbl 0593.16019)] of the model theory of modules and, initially, it was investigated using model-theoretic methods. In fact, these methods run parallel to functor-category-theoretic techniques used in, for example [\textit{L. Gruson} and \textit{C. U. Jensen}, Lect. Notes Math. 867, 234-294 (1981; Zbl 0505.18005)] and in the approach of Auslander and Reiten to the representation theory of Artin algebras. In [\textit{M. Prest}, Model theory and modules, Lond. Math. Soc. Lect. Note Ser. 130 (1988; Zbl 0634.03025)] and [\textit{C. U. Jensen} and \textit{H. Lenzing}, Model theoretic algebra, Gordon and Breach (1989; Zbl 0728.03026)] there is some combination of these techniques but it is only subsequently that the essential equivalence of these approaches has been fully realised and exploited.NEWLINENEWLINENEWLINEThus Krause is able to present a coherent and well-written account, from the functor-category perspective, of a range of results, some of which were originally proved by model theory, others using homological/functorial techniques and some proved, in formulations not obviously equivalent, independently under each approach.NEWLINENEWLINENEWLINEIt is in applications to the representation theory of finite-dimensional algebras that the Ziegler spectrum has, so far, found most success, especially through the way that certain infinite-dimensional points of the spectrum correspond to families of finite-dimensional representations (see various articles in [\textit{H. Krause} and \textit{C. M. Ringel}, Infinite length modules, Birkhäuser (2000; Zbl 0945.00021)] for more on this). Even within this area one has results which are specific to one type of algebra as well as very general results.NEWLINENEWLINENEWLINEThe monograph under review emphasises, but is by no means confined to, results of general applicability and results which apply to finite-dimensional algebras (or Artin algebras) rather than other types of ring. The author has included some of his own results, for example, those on ideals in mod-\(R\), which do not appear elsewhere.