Categories and Modules with K Theory in View

This book, first published in 2000, is a concise introduction at graduate level to ring theory, module theory and number theory.

Categories and Modules with K Theory in View

This book, first published in 2000, develops aspects of category theory fundamental to the study of algebraic K-theory. Ring and module theory illustrates category theory which provides insight into more advanced topics in module theory. Starting with categories in general, the text then examines categories of K-theory. This leads to the study of tensor products and the Morita theory. The categorical approach to localizations and completions of modules is formulated in terms of direct and inverse limits, prompting a discussion of localization of categories in general. Finally, local-global techniques which supply information about modules from their localizations and completions and underlie some interesting applications of K-theory to number theory and geometry are considered. Many useful exercises, concrete illustrations of abstract concepts placed in their historical settings and an extensive list of references are included. This book will help all who wish to work in K-theory to master its prerequisites.

An Introduction to Rings and Modules

This is a concise 2000 introduction at graduate level to ring theory, module theory and number theory.

An Introduction to Rings and Modules

This is a concise 2000 introduction at graduate level to ring theory, module theory and number theory.

Introduction to Foliations and Lie Groupoids

... number theory K. Goebel & W.A. Kirk Topics in metric fixed point theory J.F.
Humphreys Reflection groups and Coxeter ... M.E. Keating Categories and
modules with K-theory in view K. Sato Levy processes and infinitely divisible
distributions ...

Introduction to Foliations and Lie Groupoids

This book gives a quick introduction to the theory of foliations, Lie groupoids and Lie algebroids. An important feature is the emphasis on the interplay between these concepts: Lie groupoids form an indispensable tool to study the transverse structure of foliations as well as their noncommutative geometry, while the theory of foliations has immediate applications to the Lie theory of groupoids and their infinitesimal algebroids. The book starts with a detailed presentation of the main classical theorems in the theory of foliations then proceeds to Molino's theory, Lie groupoids, constructing the holonomy groupoid of a foliation and finally Lie algebroids. Among other things, the authors discuss to what extent Lie's theory for Lie groups and Lie algebras holds in the more general context of groupoids and algebroids. Based on the authors' extensive teaching experience, this book contains numerous examples and exercises making it ideal for graduate students and their instructors.

Lectures in Logic and Set Theory Volume 1 Mathematical Logic

... 40 41 42 43 K. Petersen Ergodic theory P.T. Johnstone Stone spaces J.-P.
Kahane Some random series of functions, ... M.E. Keating Categories and
modules with K-theory in view K. Sato Levy processes and infinitely divisible
distributions ...

Lectures in Logic and Set Theory  Volume 1  Mathematical Logic

This two-volume work bridges the gap between introductory expositions of logic or set theory on one hand, and the research literature on the other. It can be used as a text in an advanced undergraduate or beginning graduate course in mathematics, computer science, or philosophy. The volumes are written in a user-friendly conversational lecture style that makes them equally effective for self-study or class use. Volume 1 includes formal proof techniques, a section on applications of compactness (including nonstandard analysis), a generous dose of computability and its relation to the incompleteness phenomenon, and the first presentation of a complete proof of Godel's 2nd incompleteness since Hilbert and Bernay's Grundlagen theorem.

Mathematical Reviews

Then there is a natural equivalence of categories between the category of locally
free crystals of finite rank on Cris ( X / W ( S ) ) and the category of locally free W .
( x ) - modules of finite rank with an integrable de Rham - Witt connection
satisfying a certain nilpotence condition . ... Those who want to enter into the
realm of algebraic K - theory , with a view to arithmetic algebraic geometry , will
find in this ...

Mathematical Reviews


Galois Module Structure

This is the first published graduate course on the Chinburg conjectures, and this book provides the necessary background in algebraic and analytic number theory, cohomology, representation theory, and Hom-descriptions.

Galois Module Structure

This is the first published graduate course on the Chinburg conjectures, and this book provides the necessary background in algebraic and analytic number theory, cohomology, representation theory, and Hom-descriptions. The computation of Hom-descriptions is facilitated by Snaith's Explicit Brauer Induction technique in representation theory. In this way, illustrative special cases of the main results and new examples of the conjectures are proved and amplified by numerous exercises and research problems.

Geometric and Cohomological Group Theory

8 Proofs IV: Connective algebraic K-theory of categories of controlled simplicial
modules 227 9 Applications 239 A A simplicial mapping telescope 242. 1.
Introduction. Controlled algebra is a powerful tool to prove statements about the
algebraic K-theory of a ring R. While early on it was ... As cG(X; R) suffers from
this, we define a full subcategory point of view of like finite finite objects CW-
complexes.

Geometric and Cohomological Group Theory

Surveys the state of the art in geometric and cohomological group theory. Ideal entry point for young researchers.

Topics in Algebraic and Topological K Theory

(c) Let R be a ring and C(R) be the category of (unbounded) complexes over R.
Its objects are families of R-modules ... structures of derived categories is one of
the main objectives of dg-category theory. ... This last point of view is a bit less
precise as the original notion of localizations, as the object S−1C satisfies a ...

Topics in Algebraic and Topological K Theory

This volume is an introductory textbook to K-theory, both algebraic and topological, and to various current research topics within the field, including Kasparov's bivariant K-theory, the Baum-Connes conjecture, the comparison between algebraic and topological K-theory of topological algebras, the K-theory of schemes, and the theory of dg-categories.

Algebraic K Theory

Algebraic K Theory


K Theory

Then if X is compact, the pseudo-abelian category 'K associated with (6 is
equivalent to the category é”(X) of all vector bundles over X . Proofl Let Q be ...
Nevertheless, this point of view will sometimes be convenient for theoretical
purposes.

K Theory

From the Preface: K-theory was introduced by A. Grothendieck in his formulation of the Riemann- Roch theorem. For each projective algebraic variety, Grothendieck constructed a group from the category of coherent algebraic sheaves, and showed that it had many nice properties. Atiyah and Hirzebruch considered a topological analog defined for any compact space X, a group K{X) constructed from the category of vector bundles on X. It is this ''topological K-theory" that this book will study. Topological K-theory has become an important tool in topology. Using K- theory, Adams and Atiyah were able to give a simple proof that the only spheres which can be provided with H-space structures are S1, S3 and S7. Moreover, it is possible to derive a substantial part of stable homotopy theory from K-theory. The purpose of this book is to provide advanced students and mathematicians in other fields with the fundamental material in this subject. In addition, several applications of the type described above are included. In general we have tried to make this book self-contained, beginning with elementary concepts wherever possible; however, we assume that the reader is familiar with the basic definitions of homotopy theory: homotopy classes of maps and homotopy groups.Thus this book might be regarded as a fairly self-contained introduction to a "generalized cohomology theory".

Representations and Cohomology Volume 2 Cohomology of Groups and Modules

A further introduction to modern developments in the representation theory of finite groups and associative algebras.

Representations and Cohomology  Volume 2  Cohomology of Groups and Modules

A further introduction to modern developments in the representation theory of finite groups and associative algebras.

Sets Logic and Categories

The discussion is su pported by a wide range of exercises. The final chapter touches on philosophical issues. The book is supported by a World Wibe Web site containing a variety of supplementary material.

Sets  Logic and Categories

Set theory, logic and category theory lie at the foundations of mathematics, and have a dramatic effect on the mathematics that we do, through the Axiom of Choice, Gödel's Theorem, and the Skolem Paradox. But they are also rich mathematical theories in their own right, contributing techniques and results to working mathematicians such as the Compactness Theorem and module categories. The book is aimed at those who know some mathematics and want to know more about its building blocks. Set theory is first treated naively an axiomatic treatment is given after the basics of first-order logic have been introduced. The discussion is su pported by a wide range of exercises. The final chapter touches on philosophical issues. The book is supported by a World Wibe Web site containing a variety of supplementary material.

K theory of Forms

The description for this book, K-Theory of Forms. (AM-98), Volume 98, will be forthcoming.

K theory of Forms

The Description for this book, K-Theory of Forms. (AM-98), will be forthcoming.

Reviews in K theory 1940 84

expected form of a long exact sequence D10 QQ- AND S - CONSTRUCTION ;
ALGEBRAIC K - THEORY OF TOPOLOGICAL SPACES -K ( C ) -K ( A ) OK ( B ) -K
( R ) -K ;-( C ) -. ... For each type of generalized free product situation , certain
diagram categories of modules are introduced . For example , one fundamental
concept is that of a Mayer - Vietoris presentation of an R - module M : If Ma , Mg ,
and ...

Reviews in K theory  1940 84


Representation Theory

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 ...

Representation Theory

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.

Abstracts of Papers Presented to the American Mathematical Society

Denote by B the category of all bounded weight g - modules , i.e. those which are
direct sum of their weight spaces and ... ( Received September 19 , 2005 ) 19 K -
theory 1012-19-136 Jacques Distler ... of view is particularly powerful when the
Calabi - Yaus are not simply - connected ( and hence , there is torsion in the K ...

Abstracts of Papers Presented to the American Mathematical Society


American Book Publishing Record

A57 no. 671, QA2481 99-027220 511.3/22 21 ISBN 0-8218-1180-0 /. Set theory.
2. Forcing (Model theory) I. Shelah. Saharon II. Title. ... II. Series. BERRICK A. J. (
A. Jon) 511.3 Categories and modules with K-theory in view / A.J. Berrick, M.E. ...

American Book Publishing Record


K Theory Arithmetic and Geometry

The same result result holds for M = the category of algebraic holonomic D -
modules and D = the derived category of complexes of D - modules with
holonomic cohomology . One may look at this from two complementary points of
view .

K Theory  Arithmetic and Geometry

This volume of research papers is an outgrowth of the Manin Seminar at Moscow University, devoted to K-theory, homological algebra and algebraic geometry. The main topics discussed include additive K-theory, cyclic cohomology, mixed Hodge structures, theory of Virasoro and Neveu-Schwarz algebras.