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Author: Maurice Auslander
Publisher: American Mathematical Soc.
ISBN: 0821812947
Category : Commutative rings
Languages : en
Pages : 150
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Book Description
The notions of torsion and torsion freeness have played a very important role in module theory--particularly in the study of modules over integral domains. Furthermore, the use of homological techniques in this connection has been well established. It is the aim of this paper to extend these techniques and to show that this extension leads naturally to several new concepts (e.g. k-torsion freeness and Gorenstein dimension) which are useful in the classification of modules and rings.
Author: Maurice Auslander
Publisher: American Mathematical Soc.
ISBN: 0821812947
Category : Commutative rings
Languages : en
Pages : 150
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Book Description
The notions of torsion and torsion freeness have played a very important role in module theory--particularly in the study of modules over integral domains. Furthermore, the use of homological techniques in this connection has been well established. It is the aim of this paper to extend these techniques and to show that this extension leads naturally to several new concepts (e.g. k-torsion freeness and Gorenstein dimension) which are useful in the classification of modules and rings.
Author: Douglas C. Ravenel
Publisher: Princeton University Press
ISBN: 9780691025728
Category : Mathematics
Languages : en
Pages : 228
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Book Description
Nilpotence and Periodicity in Stable Homotopy Theory describes some major advances made in algebraic topology in recent years, centering on the nilpotence and periodicity theorems, which were conjectured by the author in 1977 and proved by Devinatz, Hopkins, and Smith in 1985. During the last ten years a number of significant advances have been made in homotopy theory, and this book fills a real need for an up-to-date text on that topic. Ravenel's first few chapters are written with a general mathematical audience in mind. They survey both the ideas that lead up to the theorems and their applications to homotopy theory. The book begins with some elementary concepts of homotopy theory that are needed to state the problem. This includes such notions as homotopy, homotopy equivalence, CW-complex, and suspension. Next the machinery of complex cobordism, Morava K-theory, and formal group laws in characteristic p are introduced. The latter portion of the book provides specialists with a coherent and rigorous account of the proofs. It includes hitherto unpublished material on the smash product and chromatic convergence theorems and on modular representations of the symmetric group.
Author: Anand Pillay
Publisher: Courier Corporation
ISBN: 0486150437
Category : Mathematics
Languages : en
Pages : 164
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Book Description
This introductory treatment covers the basic concepts and machinery of stability theory. Lemmas, corollaries, proofs, and notes assist readers in working through and understanding the material and applications. Full of examples, theorems, propositions, and problems, it is suitable for graduate students in logic and mathematics, professional mathematicians, and computer scientists. Chapter 1 introduces the notions of definable type, heir, and coheir. A discussion of stability and order follows, along with definitions of forking that follow the approach of Lascar and Poizat, plus a consideration of forking and the definability of types. Subsequent chapters examine superstability, dividing and ranks, the relation between types and sets of indiscernibles, and further properties of stable theories. The text concludes with proofs of the theorems of Morley and Baldwin-Lachlan and an extension of dimension theory that incorporates orthogonality of types in addition to regular types.
Author: Alexander Zimmermann
Publisher: Springer
ISBN: 3319079689
Category : Mathematics
Languages : en
Pages : 720
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Book Description
Introducing the representation theory of groups and finite dimensional algebras, first studying basic non-commutative ring theory, this book covers the necessary background on elementary homological algebra and representations of groups up to block theory. It further discusses vertices, defect groups, Green and Brauer correspondences and Clifford theory. Whenever possible the statements are presented in a general setting for more general algebras, such as symmetric finite dimensional algebras over a field. Then, abelian and derived categories are introduced in detail and are used to explain stable module categories, as well as derived categories and their main invariants and links between them. Group theoretical applications of these theories are given – such as the structure of blocks of cyclic defect groups – whenever appropriate. Overall, many methods from the representation theory of algebras are introduced. Representation Theory assumes only the most basic knowledge of linear algebra, groups, rings and fields and guides the reader in the use of categorical equivalences in the representation theory of groups and algebras. As the book is based on lectures, it will be accessible to any graduate student in algebra and can be used for self-study as well as for classroom use.
Author: Anthony D. Elmendorf
Publisher: American Mathematical Soc.
ISBN: 0821843036
Category : Mathematics
Languages : en
Pages : 265
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Book Description
This book introduces a new point-set level approach to stable homotopy theory that has already had many applications and promises to have a lasting impact on the subject. Given the sphere spectrum $S$, the authors construct an associative, commutative, and unital smash product in a complete and cocomplete category of ``$S$-modules'' whose derived category is equivalent to the classical stable homotopy category. This construction allows for a simple and algebraically manageable definition of ``$S$-algebras'' and ``commutative $S$-algebras'' in terms of associative, or associative and commutative, products $R\wedge SR \longrightarrow R$. These notions are essentially equivalent to the earlier notions of $A {\infty $ and $E {\infty $ ring spectra, and the older notions feed naturally into the new framework to provide plentiful examples. There is an equally simple definition of $R$-modules in terms of maps $R\wedge SM\longrightarrow M$. When $R$ is commutative, the category of $R$-modules also has a
Author: John T. Baldwin
Publisher: Cambridge University Press
ISBN: 1107168090
Category : Mathematics
Languages : en
Pages : 462
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Book Description
This book introduces first order stability theory, organized around the spectrum problem, with complete proofs of the Vaught conjecture for ω-stable theories.
Author: Idun Reiten
Publisher: American Mathematical Soc.
ISBN: 9780821808504
Category : Mathematics
Languages : en
Pages : 216
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Book Description
Surveys developments in the representation theory of finite dimensional algebras and related topics in seven papers illustrating different techniques developed over the recent years. For graduate students and researchers with a background in commutative algebra, including rings, modules, and homological algebra. Suitable as a text for an advanced graduate course. No index. Member prices are $31 for institutions and $23 for individuals, and are available to members of the Canadian Mathematical Society. Annotation copyrighted by Book News, Inc., Portland, OR
Author: David A. Craven
Publisher: Springer Nature
ISBN: 3030217922
Category : Mathematics
Languages : en
Pages : 297
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Book Description
This book provides an accessible introduction to the state of the art of representation theory of finite groups. Starting from a basic level that is summarized at the start, the book proceeds to cover topics of current research interest, including open problems and conjectures. The central themes of the book are block theory and module theory of group representations, which are comprehensively surveyed with a full bibliography. The individual chapters cover a range of topics within the subject, from blocks with cyclic defect groups to representations of symmetric groups. Assuming only modest background knowledge at the level of a first graduate course in algebra, this guidebook, intended for students taking first steps in the field, will also provide a reference for more experienced researchers. Although no proofs are included, end-of-chapter exercises make it suitable for student seminars.
Author: Hidetoshi Marubayashi
Publisher: World Scientific
ISBN: 9812818324
Category : Mathematics
Languages : en
Pages : 314
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Book Description
Ring theory has been developing through the interaction between the investigation of its own algebraic structure and its application to many areas of mathematics, computer science, and physics among others.This volume consists of a collection of survey articles by invited speakers and original articles refereed by world experts that was presented at the fifth China-Japan-Korea International Symposium. The survey articles provide some ideas of the application as well as an excellent overview of the various areas in ring theory. The original articles exhibit new ideas, tools and techniques needed for successful research investigation in ring theory and show the trend of current research.The articles cover all of the most important areas in ring theory, making this volume a useful resource book for researchers in mathematics ? both beginners and advanced experts.
Author: F. E. A. Johnson
Publisher: Cambridge University Press
ISBN: 9780521537490
Category : Mathematics
Languages : en
Pages : 282
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Book Description
This 2003 book deals with two fundamental problems in low-dimensional topology with an eye on wider context.
