An Introduction to Homological Algebra


Author: Charles A. Weibel
Publisher: Cambridge University Press
ISBN: 113964307X
Category: Mathematics
Page: N.A
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The landscape of homological algebra has evolved over the last half-century into a fundamental tool for the working mathematician. This book provides a unified account of homological algebra as it exists today. The historical connection with topology, regular local rings, and semi-simple Lie algebras are also described. This book is suitable for second or third year graduate students. The first half of the book takes as its subject the canonical topics in homological algebra: derived functors, Tor and Ext, projective dimensions and spectral sequences. Homology of group and Lie algebras illustrate these topics. Intermingled are less canonical topics, such as the derived inverse limit functor lim1, local cohomology, Galois cohomology, and affine Lie algebras. The last part of the book covers less traditional topics that are a vital part of the modern homological toolkit: simplicial methods, Hochschild and cyclic homology, derived categories and total derived functors. By making these tools more accessible, the book helps to break down the technological barrier between experts and casual users of homological algebra.

A User's Guide to Spectral Sequences


Author: John McCleary
Publisher: Cambridge University Press
ISBN: 9780521567596
Category: Mathematics
Page: 561
View: 6264

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Spectral sequences are among the most elegant and powerful methods of computation in mathematics. This book describes some of the most important examples of spectral sequences and some of their most spectacular applications. The first part treats the algebraic foundations for this sort of homological algebra, starting from informal calculations. The heart of the text is an exposition of the classical examples from homotopy theory, with chapters on the Leray-Serre spectral sequence, the Eilenberg-Moore spectral sequence, the Adams spectral sequence, and, in this new edition, the Bockstein spectral sequence. The last part of the book treats applications throughout mathematics, including the theory of knots and links, algebraic geometry, differential geometry and algebra. This is an excellent reference for students and researchers in geometry, topology, and algebra.

Cohen-Macaulay Rings


Author: Winfried Bruns,H. Jürgen Herzog
Publisher: Cambridge University Press
ISBN: 9780521566742
Category: Mathematics
Page: 453
View: 9688

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In the last two decades Cohen-Macaulay rings and modules have been central topics in commutative algebra. This book meets the need for a thorough, self-contained introduction to the homological and combinatorial aspects of the theory of Cohen-Macaulay rings, Gorenstein rings, local cohomology, and canonical modules. A separate chapter is devoted to Hilbert functions (including Macaulay's theorem) and numerical invariants derived from them. The authors emphasize the study of explicit, specific rings, making the presentation as concrete as possible. So the general theory is applied to Stanley-Reisner rings, semigroup rings, determinantal rings, and rings of invariants. Their connections with combinatorics are highlighted, e.g. Stanley's upper bound theorem or Ehrhart's reciprocity law for rational polytopes. The final chapters are devoted to Hochster's theorem on big Cohen-Macaulay modules and its applications, including Peskine-Szpiro's intersection theorem, the Evans-Griffith syzygy theorem, bounds for Bass numbers, and tight closure. Throughout each chapter the authors have supplied many examples and exercises which, combined with the expository style, will make the book very useful for graduate courses in algebra. As the only modern, broad account of the subject it will be essential reading for researchers in commutative algebra.

Commutative Ring Theory


Author: H. Matsumura,Miles Reid
Publisher: Cambridge University Press
ISBN: 9780521367646
Category: Mathematics
Page: 320
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This book explores commutative ring theory, an important a foundation for algebraic geometry and complex analytical geometry.

Local Cohomology

An Algebraic Introduction with Geometric Applications
Author: M. P. Brodmann,R. Y. Sharp
Publisher: Cambridge University Press
ISBN: 0521513634
Category: Mathematics
Page: 491
View: 5754

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This second edition of a successful graduate text provides a careful and detailed algebraic introduction to Grothendieck's local cohomology theory, including in multi-graded situations, and provides many illustrations of the theory in commutative algebra and in the geometry of quasi-affine and quasi-projective varieties. Topics covered include Serre's Affineness Criterion, the Lichtenbaum-Hartshorne Vanishing Theorem, Grothendieck's Finiteness Theorem and Faltings' Annihilator Theorem, local duality and canonical modules, the Fulton-Hansen Connectedness Theorem for projective varieties, and connections between local cohomology and both reductions of ideals and sheaf cohomology. The book is designed for graduate students who have some experience of basic commutative algebra and homological algebra and also experts in commutative algebra and algebraic geometry. Over 300 exercises are interspersed among the text; these range in difficulty from routine to challenging, and hints are provided for some of the more difficult ones.

Representations and Cohomology: Volume 1, Basic Representation Theory of Finite Groups and Associative Algebras


Author: D. J. Benson
Publisher: Cambridge University Press
ISBN: 9780521636537
Category: Mathematics
Page: 260
View: 5770

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This is the first of two volumes providing an introduction to modern developments in the representation theory of finite groups and associative algebras, which have transformed the subject into a study of categories of modules. Thus, Dr. Benson's unique perspective in this book incorporates homological algebra and the theory of representations of finite-dimensional algebras. This volume is primarily concerned with the exposition of the necessary background material, and the heart of the discussion is a lengthy introduction to the (Auslander-Reiten) representation theory of finite dimensional algebras, in which the techniques of quivers with relations and almost-split sequences are discussed in some detail.

An Introduction to Rings and Modules

With K-Theory in View
Author: A. J. Berrick,M. E. Keating
Publisher: Cambridge University Press
ISBN: 9780521632744
Category: Mathematics
Page: 265
View: 3362

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

A Course in Homological Algebra


Author: P.J. Hilton,Urs Stammbach
Publisher: Springer Science & Business Media
ISBN: 146849936X
Category: Mathematics
Page: 340
View: 5191

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In this chapter we are largely influenced in our choice of material by the demands of the rest of the book. However, we take the view that this is an opportunity for the student to grasp basic categorical notions which permeate so much of mathematics today, including, of course, algebraic topology, so that we do not allow ourselves to be rigidly restricted by our immediate objectives. A reader totally unfamiliar with category theory may find it easiest to restrict his first reading of Chapter II to Sections 1 to 6; large parts of the book are understandable with the material presented in these sections. Another reader, who had already met many examples of categorical formulations and concepts might, in fact, prefer to look at Chapter II before reading Chapter I. Of course the reader thoroughly familiar with category theory could, in principal, omit Chapter II, except perhaps to familiarize himself with the notations employed. In Chapter III we begin the proper study of homological algebra by looking in particular at the group ExtA(A, B), where A and Bare A-modules. It is shown how this group can be calculated by means of a projective presentation of A, or an injective presentation of B; and how it may also be identified with the group of equivalence classes of extensions of the quotient module A by the submodule B.

Hodge Theory and Complex Algebraic Geometry I:


Author: Claire Voisin
Publisher: Cambridge University Press
ISBN: 9781139437691
Category: Mathematics
Page: N.A
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The first of two volumes offering a modern introduction to Kaehlerian geometry and Hodge structure. The book starts with basic material on complex variables, complex manifolds, holomorphic vector bundles, sheaves and cohomology theory, the latter being treated in a more theoretical way than is usual in geometry. The author then proves the Kaehler identities, which leads to the hard Lefschetz theorem and the Hodge index theorem. The book culminates with the Hodge decomposition theorem. The meanings of these results are investigated in several directions. Completely self-contained, the book is ideal for students, while its content gives an account of Hodge theory and complex algebraic geometry as has been developed by P. Griffiths and his school, by P. Deligne, and by S. Bloch. The text is complemented by exercises which provide useful results in complex algebraic geometry.

An Introduction to Invariants and Moduli


Author: Shigeru Mukai,W. M. Oxbury
Publisher: Cambridge University Press
ISBN: 9780521809061
Category: Mathematics
Page: 503
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Incorporated in this volume are the first two books in Mukai's series on Moduli Theory. The notion of a moduli space is central to geometry. However, it's influence is not confined there; for example the theory of moduli spaces is a crucial ingredient in the proof of Fermat's last theorem. An accurate account of Mukai's influential Japanese texts, this tranlation will be a valuable resource for researchers and graduate students working in a range of areas.

Methods of Homological Algebra


Author: Sergei I. Gelfand,Yuri J. Manin
Publisher: Springer Science & Business Media
ISBN: 3662032201
Category: Mathematics
Page: 374
View: 6757

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Homological algebra first arose as a language for describing topological prospects of geometrical objects. As with every successful language it quickly expanded its coverage and semantics, and its contemporary applications are many and diverse. This modern approach to homological algebra, by two leading writers in the field, is based on the systematic use of the language and ideas of derived categories and derived functors. Relations with standard cohomology theory (sheaf cohomology, spectral sequences, etc.) are described. In most cases complete proofs are given. Basic concepts and results of homotopical algebra are also presented. The book addresses people who want to learn about a modern approach to homological algebra and to use it in their work.

Reflection Groups and Coxeter Groups


Author: James E. Humphreys
Publisher: Cambridge University Press
ISBN: 9780521436137
Category: Mathematics
Page: 204
View: 6263

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This graduate textbook presents a concrete and up-to-date introduction to the theory of Coxeter groups. The book is self-contained, making it suitable either for courses and seminars or for self-study. The first part is devoted to establishing concrete examples. Finite reflection groups acting on Euclidean spaces are discussed, and the first part ends with the construction of the affine Weyl groups, a class of Coxeter groups that plays a major role in Lie theory. The second part (which is logically independent of, but motivated by, the first) develops from scratch the properties of Coxeter groups in general, including the Bruhat ordering and the seminal work of Kazhdan and Lusztig on representations of Hecke algebras associated with Coxeter groups is introduced. Finally a number of interesting complementary topics as well as connections with Lie theory are sketched. The book concludes with an extensive bibliography on Coxeter groups and their applications.

Galois Groups and Fundamental Groups


Author: Tamás Szamuely
Publisher: Cambridge University Press
ISBN: 1139481142
Category: Mathematics
Page: N.A
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Ever since the concepts of Galois groups in algebra and fundamental groups in topology emerged during the nineteenth century, mathematicians have known of the strong analogies between the two concepts. This book presents the connection starting at an elementary level, showing how the judicious use of algebraic geometry gives access to the powerful interplay between algebra and topology that underpins much modern research in geometry and number theory. Assuming as little technical background as possible, the book starts with basic algebraic and topological concepts, but already presented from the modern viewpoint advocated by Grothendieck. This enables a systematic yet accessible development of the theories of fundamental groups of algebraic curves, fundamental groups of schemes, and Tannakian fundamental groups. The connection between fundamental groups and linear differential equations is also developed at increasing levels of generality. Key applications and recent results, for example on the inverse Galois problem, are given throughout.

Finite Group Theory


Author: M. Aschbacher
Publisher: Cambridge University Press
ISBN: 9780521786751
Category: Mathematics
Page: 304
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The book provides the basic foundations for the local theory of finite groups, the theory of classical linear groups, and the theory of buildings and BN-pairs.

Algebraic Number Theory


Author: A. Fröhlich,M. J. Taylor,Martin J. Taylor
Publisher: Cambridge University Press
ISBN: 9780521438346
Category: Mathematics
Page: 355
View: 5184

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This book provides a brisk, thorough treatment of the foundations of algebraic number theory on which it builds to introduce more advanced topics. Throughout, the authors emphasize the systematic development of techniques for the explicit calculation of the basic invariants such as rings of integers, class groups, and units, combining at each stage theory with explicit computations.

Cohomology of Groups


Author: Kenneth S. Brown
Publisher: Springer Science & Business Media
ISBN: 1468493272
Category: Mathematics
Page: 306
View: 5864

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Aimed at second year graduate students, this text introduces them to cohomology theory (involving a rich interplay between algebra and topology) with a minimum of prerequisites. No homological algebra is assumed beyond what is normally learned in a first course in algebraic topology, and the basics of the subject, as well as exercises, are given prior to discussion of more specialized topics.

Local Representation Theory

Modular Representations as an Introduction to the Local Representation Theory of Finite Groups
Author: J. L. Alperin
Publisher: Cambridge University Press
ISBN: 9780521449267
Category: Mathematics
Page: 178
View: 3632

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The aim of this text is to present some of the key results in the representation theory of finite groups. In order to keep the account reasonably elementary, so that it can be used for graduate-level courses, Professor Alperin has concentrated on local representation theory, emphasising module theory throughout. In this way many deep results can be obtained rather quickly. After two introductory chapters, the basic results of Green are proved, which in turn lead in due course to Brauer's First Main Theorem. A proof of the module form of Brauer's Second Main Theorem is then presented, followed by a discussion of Feit's work connecting maps and the Green correspondence. The work concludes with a treatment, new in part, of the Brauer-Dade theory. As a text, this book contains ample material for a one semester course. Exercises are provided at the end of most sections; the results of some are used later in the text. Representation theory is applied in number theory, combinatorics and in many areas of algebra. This book will serve as an excellent introduction to those interested in the subject itself or its applications.

Basic Category Theory


Author: Tom Leinster
Publisher: Cambridge University Press
ISBN: 1107044243
Category: Mathematics
Page: 190
View: 4480

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A short introduction ideal for students learning category theory for the first time.

Algebraic Homotopy


Author: Hans J. Baues
Publisher: Cambridge University Press
ISBN: 9780521333764
Category: Mathematics
Page: 466
View: 8895

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This book gives a general outlook on homotopy theory; fundamental concepts, such as homotopy groups and spectral sequences, are developed from a few axioms and are thus available in a broad variety of contexts. Many examples and applications in topology and algebra are discussed, including an introduction to rational homotopy theory in terms of both differential Lie algebras and De Rham algebras. The author describes powerful tools for homotopy classification problems, particularly for the classification of homotopy types and for the computation of the group homotopy equivalences. Applications and examples of such computations are given, including when the fundamental group is non-trivial. Moreover, the deep connection between the homotopy classification problems and the cohomology theory of small categories is demonstrated. The prerequisites of the book are few: elementary topology and algebra. Consequently, this account will be valuable for non-specialists and experts alike. It is an important supplement to the standard presentations of algebraic topology, homotopy theory, category theory and homological algebra.

Lectures on Logarithmic Algebraic Geometry


Author: Arthur Ogus
Publisher: Cambridge University Press
ISBN: 1107187737
Category: Mathematics
Page: 558
View: 6707

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A self-contained introduction to logarithmic geometry, a key tool for analyzing compactification and degeneration in algebraic geometry.