Introduction to Homological Algebra, 85

by Rotman

• ISBN13: 9780125992503
• ISBN10: 0125992505
• Format: Hardcover
• Copyright: 1979-06-28
• Publisher: Elsevier Science

Summary

"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." "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."--BOOK JACKET.

Author Biography

Joseph Rotman is Professor Emeritus of Mathematics at the University of Illinois at Urbana-Champaign.

Table of Contents

Preface to the Second Edition	p. x
How to Read This Book	p. xiii
Introduction
Simplicial Homology	p. 1
Categories and Functors	p. 7
Singular Homology	p. 29
Hom and Tensor
Modules	p. 37
Tensor Products	p. 69
Adjoint Isomorphisms	p. 91
Special Modules
Projective Modules	p. 98
Injective Modules	p. 115
Flat Modules	p. 131
Purity	p. 146
Specific Rings
Semisimple Rings	p. 154
von Neumann Regular Rings	p. 159
Hereditary and Dedekind Rings	p. 160
Semihereditary and Prufer Rings	p. 169
Quasi-Frobenius Rings	p. 173
Semiperfect Rings	p. 179
Localization	p. 188
Polynomial Rings	p. 203
Setting the Stage
Categorical Constructions	p. 213
Limits	p. 229
Adjoint Functor Theorem for Modules	p. 256
Sheaves	p. 273
Manifolds	p. 288
Sheaf Constructions	p. 294
Abelian Categories	p. 303
Complexes	p. 317
Homology
Homology Functors	p. 323
Derived Functors	p. 340
Left Derived Functors	p. 343
Axioms	p. 357
Covariant Right Derived Functors	p. 364
Contravariant Right Derived Functors	p. 369
Sheaf Cohomology	p. 377
Cech Cohomology	p. 384
Riemann-Roch Theorem	p. 392
Tor and Ext
Tor	p. 404
Domains	p. 412
Localization	p. 415
Ext	p. 418
Baer Sum	p. 428
Cotorsion Groups	p. 438
Universal Coefficients	p. 448
Homology and Rings
Dimensions of Rings	p. 453
Hilbert's Syzygy Theorem	p. 467
Stably Free Modules	p. 476
Commutative Noetherian Local Rings	p. 484
Homology and Groups
Group Extensions	p. 495
Semidirect Products	p. 500
General Extensions and Cohomology	p. 504
Stabilizing Automorphisms	p. 514
Group Cohomology	p. 519
Bar Resolutions	p. 525
Group Homology	p. 535
Schur Multiplier	p. 541
Change of Groups	p. 559
Restriction and Inflation	p. 564
Transfer	p. 572
Tate Groups	p. 580
Outer Automorphisms of p-Groups	p. 587
Cohomological Dimension	p. 591
Division Rings and Brauer Groups	p. 595
Spectral Sequences
Bicomplexes	p. 609
Filtrations and Exact Couples	p. 616
Convergence	p. 624
Homology of the Total Complex	p. 628
Cartan-Eilenberg Resolutions	p. 647
Grothendieck Spectral Sequences	p. 656
Groups	p. 660
Rings	p. 666
Sheaves	p. 675
Kunneth Theorems	p. 678
References	p. 689
Special Notation	p. 695
Index	p. 697
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