Infinite Length Modules

This volume presents the invited lectures of a conference devoted to Infinite Length Modules, held at Bielefeld, September 7-11, 1998. Some additional surveys have been included in order to establish a unified picture. The scientific organization of the conference was in the hands of K. Brown (Glasgow), P. M. Cohn (London), I. Reiten (Trondheim) and C. M. Ringel (Bielefeld). The conference was concerned with the role played by modules of infinite length when dealing with problems in the representation theory of algebras. The investi­ gation of such modules always relies on information concerning modules of finite length, for example simple modules and their possible extensions. But the converse is also true: recent developments in representation theory indicate that a full un­ derstanding of the category of finite dimensional modules, even over a finite dimen­ sional algebra, requires consideration of infinite dimensional, thus infinite length, modules. For instance, the important notion of tameness uses one-parameter fami­ lies of modules, or, alternatively, generic modules and they are of infinite length. If one tries to exhibit a presentation of a module category, it turns out to be essential to take into account the indecomposable modules which are algebraically compact, or, equivalently, pure injective. Specific methods have been developed over the last few years dealing with such special situations as group algebras of finite groups or noetherian rings, and there are surprising relations to topology and geometry. The conference outlined the present state of the art. Front Matter....Pages i-ix Infinite Length Modules. Some Examples as Introduction....Pages 1-73 Modules with Strange Decomposition Properties....Pages 75-87 Failure of the Krull-Schmidt Theorem for Artinian Modules and Serial Modules....Pages 89-99 Artinian Modules Over a Matrix Ring....Pages 101-105 Some Combinatorial Principles for Solving Algebraic Problems....Pages 107-127 Dimension Theory of Noetherian Rings....Pages 129-147 Krull, Gelfand-Kirillov, Filter, Faithful and Schur Dimensions....Pages 149-166 Cohen — Macaulay Modules and Approximations....Pages 167-192 The Generic Representation Theory of Finite Fields: A Survey of Basic Structure....Pages 193-212 Unstable Modules Over the Steenrod Algebra, Functors, and the Cohomology of Spaces....Pages 213-228 Infinite Dimensional Modules for Finite Groups....Pages 229-249 Bousfield Localization for Representation Theorists....Pages 251-272 The Thick Subcategory Generated by the Trivial Module....Pages 273-283 Birational Classification of Moduli Spaces....Pages 285-296 Tame Algebras and Degenerations of Modules....Pages 297-309 On Some Tame and Discrete Families of Modules....Pages 311-319 Purity, Algebraic Compactness, Direct Sum Decompositions, and Representation Type....Pages 321-330 Topological and Geometric Aspects of the Ziegler Spectrum....Pages 331-367 Finite Versus Infinite Dimensional Representations — A New Definition of Tameness....Pages 369-392 Invariance of Tameness under Stable Equivalence: Krause’s Theorem....Pages 393-403 The Krull-Gabriel Dimension of an Algebra — Open Problems and Conjectures....Pages 405-418 Homological Differences between Finite and Infinite Dimensional Representations of Algebras....Pages 419-424 ....Pages 425-439 This volume presents the invited lectures of a conference devoted to Infinite Length Modules, held at Bielefeld, September 7-11, 1998. Some additional surveys have been included in order to establish a unified picture. The scientific organization of the conference was in the hands of K. Brown (Glasgow), P.M. Cohn (London), I. Reiten (Trondheim) and C.M. Ringel (Bielefeld). The conference was concerned with the role played by modules of infinite length when dealing with problems in the representation theory of algebras. The investiƯ gation of such modules always relies on information concerning modules of finite length, for example simple modules and their possible extensions. But the converse is also true: recent developments in representation theory indicate that a full unƯ derstanding of the category of finite dimensional modules, even over a finite dimenƯ sional algebra, requires consideration of infinite dimensional, thus infinite length, modules. For instance, the important notion of tameness uses one-parameter famiƯ lies of modules, or, alternatively, generic modules and they are of infinite length. If one tries to exhibit a presentation of a module category, it turns out to be essential to take into account the indecomposable modules which are algebraically compact, or, equivalently, pure injective. Specific methods have been developed over the last few years dealing with such special situations as group algebras of finite groups or noetherian rings, and there are surprising relations to topology and geometry. The conference outlined the present state of the art This book is concerned with the role played by modules of infinite length when dealing with problems in the representation theory of groups and algebras, but also in topology and geometry; it shows the intriguing interplay between finite and infinite length modules. The volume presents the invited lectures of a conference devoted to Infinite Length Modules, held at Bielefeld in September 1998, which brought together experts from quite different schools in order to survey surprising relations between algebra, topology and geometry. Some additional reports have been included in order to establish a unified picture. The collection of articles, written by well-known experts from all parts of the world, is conceived as a sort of handbook which provides an easy access to the present state of knowledge and its aim is to stimulate further development. edited by Henning Krause, Claus Michael Ringel. Concerned with the role played by modules of infinite length when dealing with problems in the representation theory of groups and algebras, but also in topology and geometry, this book shows the intriguing interplay between finite and infinite length modules. It presents the invited lectures of a conference devoted to 'Infinite Length Modules'.