An introduction to homological algebra

Joseph J. Rotman9780387245270, 0387245278

With a wealth of examples as well as abundant applications to algebra, this is a must-read work: an easy-to-follow, step-by-step 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.

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Table of contents :
front-matter.pdf……Page 1
Contents……Page 5
(Preface to the Second Edition!!)……Page 8
(How to Read This Book!!)……Page 11
(Simplicial Homology!!1.1)……Page 13
(Categories and Functors!!1.2)……Page 19
(Singular Homology!!1.3)……Page 40
(Modules!!2.1)……Page 49
(Tensor Products!!2.2)……Page 81
(Adjoint Isomorphisms!!2.2.1)……Page 103
(Projective Modules!!3.1)……Page 110
(Injective Modules!!3.2)……Page 127
(Flat Modules!!3.3)……Page 143
(Purity!!3.3.1)……Page 158
(Semisimple Rings!!4.1)……Page 166
(von Neumann Regular Rings!!4.2)……Page 171
(Hereditary and Dedekind Rings!!4.3)……Page 172
(Semihereditary and Prüfer Rings!!4.4)……Page 181
(Quasi-Frobenius Rings!!4.5)……Page 185
(Semiperfect Rings!!4.6)……Page 191
(Localization!!4.7)……Page 200
(Polynomial Rings!!4.8)……Page 214
(Categorical Constructions!!5.1)……Page 225
(Limits!!5.2)……Page 241
(Adjoint Functor Theorem for Modules!!5.3)……Page 268
(Sheaves!!5.4)……Page 285
(Manifolds!!5.4.1)……Page 300
(Sheaf Constructions!!5.4.2)……Page 306
(Abelian Categories!!5.5)……Page 315
(Complexes!!5.5.1)……Page 329
(Homology Functors!!6.1)……Page 335
(Derived Functors!!6.2)……Page 352
(Left Derived Functors!!6.2.1)……Page 355
(Axioms!!6.2.2)……Page 368
(Covariant Right Derived Functors!!6.2.3)……Page 375
(Contravariant Right Derived Functors!!6.2.4)……Page 381
(Sheaf Cohomology!!6.3)……Page 389
(Cech Cohomology!!6.3.1)……Page 395
(Riemann–Roch Theorem!!6.3.2)……Page 404
(Tor!!7.1)……Page 416
(Domains!!7.1.1)……Page 424
(Localization!!7.1.2)……Page 426
(Ext!!7.2)……Page 429
(Baer Sum!!7.2.1)……Page 440
(Cotorsion Groups!!7.3)……Page 449
(Universal Coefficients!!7.4)……Page 459
(Dimensions of Rings!!8.1)……Page 465
(Hilbert’s Syzygy Theorem!!8.2)……Page 479
(Stably Free Modules!!8.3)……Page 488
(Commutative Noetherian Local Rings!!8.4)……Page 496
(Group Extensions!!9.1)……Page 507
(Semidirect Products!!9.1.1)……Page 511
(General Extensions and Cohomology!!9.1.2)……Page 516
(Stabilizing Automorphisms!!9.1.3)……Page 526
(Group Cohomology!!9.2)……Page 530
(Bar Resolutions!!9.3)……Page 537
(Group Homology!!9.4)……Page 547
(Schur Multiplier!!9.4.1)……Page 553
(Change of Groups!!9.5)……Page 571
(Restriction and Inflation!!9.5.1)……Page 576
(Transfer!!9.6)……Page 583
(Tate Groups!!9.7)……Page 592
(Outer Automorphisms of p-Groups!!9.8)……Page 598
(Cohomological Dimension!!9.9)……Page 603
(Division Rings and Brauer Groups!!9.10)……Page 607
(Bicomplexes!!10.1)……Page 620
(Filtrations and Exact Couples!!10.2)……Page 627
(Convergence!!10.3)……Page 635
(Homology of the Total Complex!!10.4)……Page 639
(Cartan–Eilenberg Resolutions!!10.5)……Page 659
(Grothendieck Spectral Sequences!!10.6)……Page 667
(Groups!!10.7)……Page 672
(Rings!!10.8)……Page 678
(Sheaves!!10.9)……Page 687
(Künneth Theorems!!10.10)……Page 690
References……Page 701
Special Notation……Page 707
Index……Page 709