An Introduction to Homological Algebra

 
9780123994660: An Introduction to Homological Algebra

An Introduction to Homological Algebra discusses the origins of algebraic topology. It also presents the study of homological algebra as a two-stage affair. First, one must learn the language of Ext and Tor and what it describes. Second, one must be able to compute these things, and often, this involves yet another language: spectral sequences. Homological algebra is an accessible subject to those who wish to learn it, and this book is the author’s attempt to make it lovable.
This book comprises 11 chapters, with an introductory chapter that focuses on line integrals and independence of path, categories and functors, tensor products, and singular homology. Succeeding chapters discuss Hom and ?; projectives, injectives, and flats; specific rings; extensions of groups; homology; Ext; Tor; son of specific rings; the return of cohomology of groups; and spectral sequences, such as bicomplexes, Kunneth Theorems, and Grothendieck Spectral Sequences.
This book will be of interest to practitioners in the field of pure and applied mathematics.

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From the Back Cover:

With a wealth of examples as well as abundant applications to Algebra, this is a must-read work: a clearly written, easy-to-follow guide to Homological Algebra.  The author provides a treatment of Homological Algebra which approaches the subject in terms of its origins in algebraic topology.  In this brand new edition the text has been fully updated and revised throughout and new material on sheaves and abelian categories has been added.

 

Applications include the following:

 

* to rings -- Lazard's theorem that flat modules are direct limits of free modules, Hilbert's Syzygy Theorem, Quillen-Suslin's solution of Serre's problem about projectives over polynomial rings, Serre-Auslander-Buchsbaum characterization of regular local rings (and a sketch of unique factorization);

 

* to groups -- Schur-Zassenhaus, Gaschutz's theorem on outer automorphisms of finite p-groups, Schur multiplier, cotorsion groups;

 

* to sheaves -- sheaf cohomology, Cech cohomology, discussion of Riemann-Roch Theorem over compact Riemann surfaces.

 

Learning Homological Algebra is a two-stage affair. Firstly, one must learn the language of Ext and Tor, and what this describes. Secondly, one must be able to compute these things using a separate language: that of spectral sequences. The basic properties of spectral sequences are developed using exact couples. All is done in the context of bicomplexes, for almost all applications of spectral sequences involve indices.  Applications include Grothendieck spectral sequences, change of rings, Lyndon-Hochschild-Serre sequence, and theorems of Leray and Cartan computing sheaf cohomology.

 

Joseph Rotman is Professor Emeritus of Mathematics at the University of Illinois at Urbana-Champaign. He is the author of numerous successful textbooks, including Advanced Modern Algebra (Prentice-Hall 2002), Galois Theory, 2nd Edition (Springer 1998) A First Course in Abstract Algebra (Prentice-Hall 1996), Introduction to the Theory of Groups, 4th Edition (Springer 1995), and Introduction to Algebraic Topology (Springer 1988).

Review:

From the reviews of the second edition:

"Joseph J. Rotman is a renowned textbook author in contemporary mathematics. Over the past four decades, he has published numerous successful texts of introductory character, mainly in the field of modern abstract algebra and its related disciplines. ... Now, in the current second edition, the author has reworked the original text considerably.  While the first edition covered exclusively aspects of the homological algebra of groups, rings, and modules, that is, topics from its first period of development, the new edition includes some additional material from the second period, together with numerous other, more recent results from the homological algebra of groups, rings, and modules. The new edition has almost doubled in size and represents a substantial updating of the classic original. ... All together, a popular classic has been turned into a new, much more topical and comprehensive textbook on homological algebra, with all the great features that once distinguished the original, very much to the belief [of its] new generation of readers." (Werner Kleinert, Zentralblatt)

"The new expanded second edition ... attempts to cover more ground, basically going from the (concrete) category of modules over a given ring, as in the first edition, to an abelian category and to treat the important example of the category of sheaves on a topological space. ... the exercise at the end of every section, plenty of examples and motivation for the many new concepts set this book apart and make it an ideal textbook for a course on the subject." (Felipe Zaldivar, MAA Online, December, 2008)

"This is the second edition of Rotman’s introduction to the more classical aspects of homological algebra ... . The book is mainly concerned with homological algebra in module categories ... . The book is full of illustrative examples and exercises. It contains many references for further study and also to original sources. All this makes Rotman’s book very convenient for beginners in homological algebra as well as a reference book." (Fernando Muro, Mathematical Reviews, Issue 2009 i)

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Rotman, Joseph J.
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Joseph J. Rotman
Published by Academic Press (1979)
ISBN 10: 0123994667 ISBN 13: 9780123994660
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