An Introduction to Homological Algebra (Paperback)

Joseph J. Rotman

ISBN 10: 0387245278 / ISBN 13: 9780387245270

Editore: Springer-Verlag New York Inc., New York, NY, 2008

Riguardo questo articolo

Paperback. Learning homological algebra is 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. This book gives a treatment of homological algebra which motivates the subject in terms of its origins in algebraic topology.In this new edition the book has been updated and revised throughout and new material on sheaves and cup products has been added. The author also included material about homotopical algebra, alias K-theory, contrasting it with homological algebra. Homological Algebra has grown in the nearly three decades since the rst e- tion of this book appeared in 1979. In their Foreword, Gelfand and Manin divide the history of Homological Algebra into three periods: the rst period ended in the early 1960s, culminating in applications of Ho- logical Algebra to regular local rings. Shipping may be from multiple locations in the US or from the UK, depending on stock availability.

Codice articolo 9780387245270

Dati bibliografici

• Titolo: An Introduction to Homological Algebra (Paperback)
• Editore: Springer-Verlag New York Inc., New York, NY
• Data di pubblicazione: 2008
• Lingua: Inglese
• ISBN 10: 0387245278
• ISBN 13: 9780387245270
• Rilegatura: Paperback
• Edizione: seconda edizione
• Condizione: new

Descrizione articolo

Riassunto

Learning homological algebra is 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. This book gives a treatment of homological algebra which motivates the subject in terms of its origins in algebraic topology.In this new edition the book has been updated and revised throughout and new material on sheaves and cup products has been added. The author also included material about homotopical algebra, alias K-theory, contrasting it with homological algebra.

Dalla quarta di copertina

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).