Infinite Length Modules

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Infinite Length Modules

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Author : Henning Krause
language : en
Publisher: Birkhäuser
Release Date : 2012-10-29

Infinite Length Modules written by Henning Krause and has been published by Birkhäuser this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-10-29 with Mathematics categories.

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.

Infinite Length Modules

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Author : Henning Krause
language : en
Publisher: Birkhäuser
Release Date : 2012-12-06

Infinite Length Modules written by Henning Krause and has been published by Birkhäuser this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.

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.

Models Modules And Abelian Groups

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Author : Rüdiger Göbel
language : en
Publisher: Walter de Gruyter
Release Date : 2008-12-10

Models Modules And Abelian Groups written by Rüdiger Göbel and has been published by Walter de Gruyter this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008-12-10 with Mathematics categories.

This is a memorial volume dedicated to A. L. S. Corner, previously Professor in Oxford, who published important results on algebra, especially on the connections of modules with endomorphism algebras. The volume contains refereed contributions which are related to the work of Corner. It contains also an unpublished extended paper of Corner himself. A memorial volume with important contributions related to algebra.

Trends In Representation Theory Of Algebras And Related Topics

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Author : José Antonio de la Peña
language : en
Publisher: American Mathematical Soc.
Release Date : 2006

Trends In Representation Theory Of Algebras And Related Topics written by José Antonio de la Peña and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006 with Mathematics categories.

This book is based on lectures given during a Workshop on Representations of Algebras and Related Topics. Some additional articles are included in order to complete a panoramic view of the main trends of the subject. The volume contains original presentations by leading algebraists addressed to specialists as well as to a broader mathematical audience. The articles include new proofs, examples, and detailed arguments. Topics under discussion include moduli spaces associated to quivers, canonical basis of quantum algebras, categorifications and derived categories, $A$-infinity algebras and functor categories, cluster algebras, support varieties for modules and complexes, the Gabriel-Roiter measure for modules, and selfinjective algebras.

Module Theory

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Author : Alberto Facchini
language : en
Publisher: Springer Science & Business Media
Release Date : 1998-06-16

Module Theory written by Alberto Facchini and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 1998-06-16 with Mathematics categories.

This expository monograph was written for three reasons. Firstly, we wanted to present the solution to a problem posed by Wolfgang Krull in 1932 [Krull 32]. He asked whether what we now call the "Krull-Schmidt Theorem" holds for ar tinian modules. The problem remained open for 63 years: its solution, a negative answer to Krull's question, was published only in 1995 (see [Facchini, Herbera, Levy and Vamos]). Secondly, we wanted to present the answer to a question posed by Warfield in 1975 [Warfield 75]. He proved that every finitely pre sented module over a serial ring is a direct sum of uniserial modules, and asked if such a decomposition was unique. In other words, Warfield asked whether the "Krull-Schmidt Theorem" holds for serial modules. The solution to this problem, a negative answer again, appeared in [Facchini 96]. Thirdly, the so lution to Warfield's problem shows interesting behavior, a rare phenomenon in the history of Krull-Schmidt type theorems. Essentially, the Krull-Schmidt Theorem holds for some classes of modules and not for others. When it does hold, any two indecomposable decompositions are uniquely determined up to a permutation, and when it does not hold for a class of modules, this is proved via an example. For serial modules the Krull-Schmidt Theorem does not hold, but any two indecomposable decompositions are uniquely determined up to two permutations. We wanted to present such a phenomenon to a wider math ematical audience.

Model Theory And Modules

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Author : Mike Prest
language : en
Publisher: Cambridge University Press
Release Date : 1988-02-25

Model Theory And Modules written by Mike Prest and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1988-02-25 with Mathematics categories.

In recent years the interplay between model theory and other branches of mathematics has led to many deep and intriguing results. In this, the first book on the topic, the theme is the interplay between model theory and the theory of modules. The book is intended to be a self-contained introduction to the subject and introduces the requisite model theory and module theory as it is needed. Dr Prest develops the basic ideas concerning what can be said about modules using the information which may be expressed in a first-order language. Later chapters discuss stability-theoretic aspects of modules, and structure and classification theorems over various types of rings and for certain classes of modules. Both algebraists and logicians will enjoy this account of an area in which algebra and model theory interact in a significant way. The book includes numerous examples and exercises and consequently will make an ideal introduction for graduate students coming to this subject for the first time.

Advances In Algebra And Model Theory

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Author : M Droste
language : en
Publisher: CRC Press
Release Date : 2019-08-16

Advances In Algebra And Model Theory written by M Droste and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-08-16 with Mathematics categories.

Contains 25 surveys in algebra and model theory, all written by leading experts in the field. The surveys are based around talks given at conferences held in Essen, 1994, and Dresden, 1995. Each contribution is written in such a way as to highlight the ideas that were discussed at the conferences, and also to stimulate open research problems in a form accessible to the whole mathematical community. The topics include field and ring theory as well as groups, ordered algebraic structure and their relationship to model theory. Several papers deal with infinite permutation groups, abelian groups, modules and their relatives and representations. Model theoretic aspects include quantifier elimination in skew fields, Hilbert's 17th problem, (aleph-0)-categorical structures and Boolean algebras. Moreover symmetry questions and automorphism groups of orders are covered. This work contains 25 surveys in algebra and model theory, each is written in such a way as to highlight the ideas that were discussed at Conferences, and also to stimulate open research problems in a form accessible to the whole mathematical community.

Classical Lie Algebras At Infinity

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Author : Ivan Penkov
language : en
Publisher: Springer Nature
Release Date : 2022-01-05

Classical Lie Algebras At Infinity written by Ivan Penkov and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-01-05 with Mathematics categories.

Originating from graduate topics courses given by the first author, this book functions as a unique text-monograph hybrid that bridges a traditional graduate course to research level representation theory. The exposition includes an introduction to the subject, some highlights of the theory and recent results in the field, and is therefore appropriate for advanced graduate students entering the field as well as research mathematicians wishing to expand their knowledge. The mathematical background required varies from chapter to chapter, but a standard course on Lie algebras and their representations, along with some knowledge of homological algebra, is necessary. Basic algebraic geometry and sheaf cohomology are needed for Chapter 10. Exercises of various levels of difficulty are interlaced throughout the text to add depth to topical comprehension. The unifying theme of this book is the structure and representation theory of infinite-dimensional locally reductive Lie algebras and superalgebras. Chapters 1-6 are foundational; each of the last 4 chapters presents a self-contained study of a specialized topic within the larger field. Lie superalgebras and flag supermanifolds are discussed in Chapters 3, 7, and 10, and may be skipped by the reader.

Lectures On Sl 2 C Modules

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Author :
language : en
Publisher: World Scientific
Release Date : 2009

Lectures On Sl 2 C Modules written by and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009 with categories.

Rings And Categories Of Modules

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Author : F.W. Anderson
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06

Rings And Categories Of Modules written by F.W. Anderson and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.

This book is intended to provide a reasonably self-contained account of a major portion of the general theory of rings and modules suitable as a text for introductory and more advanced graduate courses. We assume the famil iarity with rings usually acquired in standard undergraduate algebra courses. Our general approach is categorical rather than arithmetical. The continuing theme of the text is the study of the relationship between the one-sided ideal structure that a ring may possess and the behavior of its categories of modules. Following a brief outline of set-theoretic and categorical foundations, the text begins with the basic definitions and properties of rings, modules and homomorphisms and ranges through comprehensive treatments of direct sums, finiteness conditions, the Wedderburn-Art in Theorem, the Jacobson radical, the hom and tensor functions, Morita equivalence and duality, de composition theory of injective and projective modules, and semiperfect and perfect rings. Both to illustrate the text and to extend it we have included a substantial number of exercises covering a wide spectrum of difficulty. There are, of course, many important areas of ring and module theory that the text does not touch upon. For example, we have made no attempt to cover such subjects as homology, rings of quotients, or commutative ring theory.