Simply put, this book could have some real purpose for someone wanting a gentle introduction into homological algebra if not for one huge blunder. Rotman does do a good job at motivating a lot of the topics and not becoming too longwinded, but there is also an unfortunate fatal flaw in this book as well. This book contains far too many errors to be acceptable. While I acknowledge that all books will contain errors, this amount is beyond a level that should have been allowed to be printed without correction. Some are simple typos that will not affect the average reader. Others, however, will make this book not cater well to its target audience. The pace of this book is too slow as to make it a necessary resource, as Weibel's book of the same title or Kenneth Brown's "Cohomology of Groups" are far more rigorous and complete. This books aim seems to be aimed thus at the graduate level or possibly a mathematician from a different field. However, this audience will be in for a chore. Many mistakes lead to incorrect proofs and even worse incorrect proposition and theorem statements. When trying to understand the functorality of certain constructions, for instance, it is crucial that the reader understand exactly how things work. The mixing up of rings and modules often leaves statements paradoxical. The advanced reader will have no problem finding and fixing these errors, but for those not comfortable in this area of mathematics, this may be a huge challenge. This book may be helpful to some as a secondary resource as it does work out some simpler results that many books (e.g. the ones mentioned above) take for granted. I would not recommend this book for any other reason though.
I will be fair and say that if this book were to receive a major editing job removing most of the errors that it could be a very useful introduction. However, until such a revision is produced, buy a better book.
Author(s): Joseph J. Rotman
Series: Universitext
Edition: 2nd
Publisher: Springer
Year: 2008
Language: English
Pages: 722
front-matter.pdf......Page 2
Contents......Page 6
(Preface to the Second Edition!!)......Page 9
(How to Read This Book!!)......Page 12
(Simplicial Homology!!1.1)......Page 14
(Categories and Functors!!1.2)......Page 20
(Singular Homology!!1.3)......Page 41
(Modules!!2.1)......Page 50
(Tensor Products!!2.2)......Page 82
(Adjoint Isomorphisms!!2.2.1)......Page 104
(Projective Modules!!3.1)......Page 111
(Injective Modules!!3.2)......Page 128
(Flat Modules!!3.3)......Page 144
(Purity!!3.3.1)......Page 159
(Semisimple Rings!!4.1)......Page 167
(von Neumann Regular Rings!!4.2)......Page 172
(Hereditary and Dedekind Rings!!4.3)......Page 173
(Semihereditary and Prüfer Rings!!4.4)......Page 182
(Quasi-Frobenius Rings!!4.5)......Page 186
(Semiperfect Rings!!4.6)......Page 192
(Localization!!4.7)......Page 201
(Polynomial Rings!!4.8)......Page 215
(Categorical Constructions!!5.1)......Page 226
(Limits!!5.2)......Page 242
(Adjoint Functor Theorem for Modules!!5.3)......Page 269
(Sheaves!!5.4)......Page 286
(Manifolds!!5.4.1)......Page 301
(Sheaf Constructions!!5.4.2)......Page 307
(Abelian Categories!!5.5)......Page 316
(Complexes!!5.5.1)......Page 330
(Homology Functors!!6.1)......Page 336
(Derived Functors!!6.2)......Page 353
(Left Derived Functors!!6.2.1)......Page 356
(Axioms!!6.2.2)......Page 369
(Covariant Right Derived Functors!!6.2.3)......Page 376
(Contravariant Right Derived Functors!!6.2.4)......Page 382
(Sheaf Cohomology!!6.3)......Page 390
(Cech Cohomology!!6.3.1)......Page 396
(Riemann--Roch Theorem!!6.3.2)......Page 405
(Tor!!7.1)......Page 417
(Domains!!7.1.1)......Page 425
(Localization!!7.1.2)......Page 427
(Ext!!7.2)......Page 430
(Baer Sum!!7.2.1)......Page 441
(Cotorsion Groups!!7.3)......Page 450
(Universal Coefficients!!7.4)......Page 460
(Dimensions of Rings!!8.1)......Page 466
(Hilbert's Syzygy Theorem!!8.2)......Page 480
(Stably Free Modules!!8.3)......Page 489
(Commutative Noetherian Local Rings!!8.4)......Page 497
(Group Extensions!!9.1)......Page 508
(Semidirect Products!!9.1.1)......Page 512
(General Extensions and Cohomology!!9.1.2)......Page 517
(Stabilizing Automorphisms!!9.1.3)......Page 527
(Group Cohomology!!9.2)......Page 531
(Bar Resolutions!!9.3)......Page 538
(Group Homology!!9.4)......Page 548
(Schur Multiplier!!9.4.1)......Page 554
(Change of Groups!!9.5)......Page 572
(Restriction and Inflation!!9.5.1)......Page 577
(Transfer!!9.6)......Page 584
(Tate Groups!!9.7)......Page 593
(Outer Automorphisms of p-Groups!!9.8)......Page 599
(Cohomological Dimension!!9.9)......Page 604
(Division Rings and Brauer Groups!!9.10)......Page 608
(Bicomplexes!!10.1)......Page 621
(Filtrations and Exact Couples!!10.2)......Page 628
(Convergence!!10.3)......Page 636
(Homology of the Total Complex!!10.4)......Page 640
(Cartan--Eilenberg Resolutions!!10.5)......Page 660
(Grothendieck Spectral Sequences!!10.6)......Page 668
(Groups!!10.7)......Page 673
(Rings!!10.8)......Page 679
(Sheaves!!10.9)......Page 688
(Künneth Theorems!!10.10)......Page 691
References......Page 702
Special Notation......Page 708
Index......Page 710
