##
**An introduction to homological algebra.
2nd ed.**
*(English)*
Zbl 1157.18001

Universitext. Berlin: Springer (ISBN 978-0-387-24527-0/pbk; 978-0-387-68324-9/ebook). xiv, 709 p. (2009).

Joseph J. Rotman is a renowned textbook author in contemporary mathemathematics. 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. Standing out by their particular lucidity, comprehensiveness, didactic mastery, and widespread popularity, many of J. Rotman’s books underwent several new editions and became standard classics. In this remarkable series of great textbooks, his primer “An introduction to homological algebra” [Pure Appl. Math. 85 (1979; Zbl 0441.18018)] has played an exceptionally important role.

In fact, first published in 1979 as an expanded version of the author’s earlier booklet “Notes on homological algebra” from Van Nostrand Reinhold Math. Stud. 27 (1970; Zbl 0222.18003), this book was one of the very first systematic introductions to the subject for students, apart from its three famous predecessors by Cartan-Eilenberg (1956), Northcott (1960), and Mac Lane (1963). Moreover, Rotman’s book on homological algebra was primarily geared toward (graduate) students, much more than the others mentioned above, and the author had set high value on developing the very general, virtually hyperabstract framework of homological algebra in as vivid, comprehensible and appealing a manner as any possible. As for the details concerning both the contents and the style of the well-tried first edition of Rotman’s classic “Introduction to homological algebra”, we may unrestrictedly refer to the excellent and exhaustive review of it by J. Weinstein from thirty years ago [loc. cit.].

The book under review is the second, revised and substantially enlarged edition of Rotman’s standard text on homological algebra. In the course of the past three decades, homological algebra has undergone a tremendous development with regard to its contents, its basics, its ubiquity as a fundamental toolkit in various areas of contemporary mathematics, and its applications to mathematical physics. Originated from algebraic topology in the 1940s, homological algebra was first developed in the context of (local) commutative algebra until about 1960, then extended to abelian categories, sheaf theory, and algebraic geometry by Grothendieck and Serre through the following twenty years, and finally generalized to a rather independent discipline involving derived categories, triangulated categories, and various abstract cohomology theories. This third historical period of homological algebra is still rapidly ongoing. In the meantime, some textbooks reflecting all these aspects have appeared, for instance the books by C. Weibel [“Introduction to homological algebra”, Cambridge: Stud. Adv. Math. 38 (1994; Zbl 0797.18001)] and by S. I. Gelfand and Yu. I. Manin [“Methods of homological algebra”, Berlin: Springer (2003; Zbl 1006.18001)], but Rotman’s classic primer has nevertheless maintained both its didactic value and its popularity.

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. In the course of this major revision, the author has largely rearranged and polished the original parts of the text, and organically woven in the wealth of new and additional material, while keeping the well-proven didactic style of exposition intact throughout the entire new edition of the book.

As for the precise contents, the current new edition comes with ten chapters, whilst the original consisted of eleven chapters. However, in the new disposition, the single chapters appear as much more comprehensive than the ones in the original edition of the book, and the material has been organized in a much more systematic and efficient way, thereby reflecting once more the author’s expository mastery. Chapter 1 introduces simplicial homology, singular homology, and the language of categories and functors both as motivation and as prerequisites for the following. Chapter 2 provides the basics of module theory, including the Horn functors, tensor products, and their adjointness. Chapter 3 discusses projective, injective, and flat modules as basic concepts in “elementary” homological algebra, whereas Chapter 4 is devoted to specific classes of rings and their properties. Here the author touches upon semisimple rings, von Neumann regular rings, hereditary and Dedekind rings, Prüfer rings, quasi-Frobenius rings, semiperfect rings, the homological algebra of polynomial rings, and the localization principle. Chapter 5 is, as the author puts it, setting the stage for abstract homological algebra, and that by explaining categorical constructions, limits in categories, adjoint functors, the basics of sheaf theory, real manifolds and differential forms, abelian categories, and complexes in abelian categories. Abstract homology functors are the subject of Chapter 6, with special emphasis on derived functors, Ext and Tor, sheaf cohomology via Godement resolutions, Čech cohomology, and on the Riemann-Roch theorem for compact Riemann surfaces. Chapter 7 analyzes the functors Ext and Tor more closely, including their exact sequences, Baer sums, cotorsion groups, and the universal coefficient theorem for homology in its different versions. Chapter 8 turns to the more advanced homological algebra of rings and modules. The reader meets here the basic dimension theory for rings and modules, Hilbert’s syzygy theorem, stably free modules and Serre’s theorem on finitely generated projective modules over polynomial rings, and the theorem of Auslander-Buchsbaum-Serre on regular local noetherian rings. Chapter 9 gives an ample treatment of the homological aspects of group theory, including group extensions, group cohomology, Schur multipliers, Tate groups, outer automorphisms of finite \(p\)-groups, cohomological dimension of groups, division rings, and Brauer groups. The final Chapter 10 developes the machinery of spectral sequences via exact couples, with many applications to groups, rings, sheaves, and Künneth theorems in (co)homology.

Apart from the many enhancements and improvements, the current new edition comes with an updated bibliography and with an additional index of notation. Also, there is a much larger number of instructive examples and exercises in comparison to the first edition of the book, and the author has added a section titled “How to read this book” for the convenience of the reader. 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 it of new generations of readers.

In fact, first published in 1979 as an expanded version of the author’s earlier booklet “Notes on homological algebra” from Van Nostrand Reinhold Math. Stud. 27 (1970; Zbl 0222.18003), this book was one of the very first systematic introductions to the subject for students, apart from its three famous predecessors by Cartan-Eilenberg (1956), Northcott (1960), and Mac Lane (1963). Moreover, Rotman’s book on homological algebra was primarily geared toward (graduate) students, much more than the others mentioned above, and the author had set high value on developing the very general, virtually hyperabstract framework of homological algebra in as vivid, comprehensible and appealing a manner as any possible. As for the details concerning both the contents and the style of the well-tried first edition of Rotman’s classic “Introduction to homological algebra”, we may unrestrictedly refer to the excellent and exhaustive review of it by J. Weinstein from thirty years ago [loc. cit.].

The book under review is the second, revised and substantially enlarged edition of Rotman’s standard text on homological algebra. In the course of the past three decades, homological algebra has undergone a tremendous development with regard to its contents, its basics, its ubiquity as a fundamental toolkit in various areas of contemporary mathematics, and its applications to mathematical physics. Originated from algebraic topology in the 1940s, homological algebra was first developed in the context of (local) commutative algebra until about 1960, then extended to abelian categories, sheaf theory, and algebraic geometry by Grothendieck and Serre through the following twenty years, and finally generalized to a rather independent discipline involving derived categories, triangulated categories, and various abstract cohomology theories. This third historical period of homological algebra is still rapidly ongoing. In the meantime, some textbooks reflecting all these aspects have appeared, for instance the books by C. Weibel [“Introduction to homological algebra”, Cambridge: Stud. Adv. Math. 38 (1994; Zbl 0797.18001)] and by S. I. Gelfand and Yu. I. Manin [“Methods of homological algebra”, Berlin: Springer (2003; Zbl 1006.18001)], but Rotman’s classic primer has nevertheless maintained both its didactic value and its popularity.

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. In the course of this major revision, the author has largely rearranged and polished the original parts of the text, and organically woven in the wealth of new and additional material, while keeping the well-proven didactic style of exposition intact throughout the entire new edition of the book.

As for the precise contents, the current new edition comes with ten chapters, whilst the original consisted of eleven chapters. However, in the new disposition, the single chapters appear as much more comprehensive than the ones in the original edition of the book, and the material has been organized in a much more systematic and efficient way, thereby reflecting once more the author’s expository mastery. Chapter 1 introduces simplicial homology, singular homology, and the language of categories and functors both as motivation and as prerequisites for the following. Chapter 2 provides the basics of module theory, including the Horn functors, tensor products, and their adjointness. Chapter 3 discusses projective, injective, and flat modules as basic concepts in “elementary” homological algebra, whereas Chapter 4 is devoted to specific classes of rings and their properties. Here the author touches upon semisimple rings, von Neumann regular rings, hereditary and Dedekind rings, Prüfer rings, quasi-Frobenius rings, semiperfect rings, the homological algebra of polynomial rings, and the localization principle. Chapter 5 is, as the author puts it, setting the stage for abstract homological algebra, and that by explaining categorical constructions, limits in categories, adjoint functors, the basics of sheaf theory, real manifolds and differential forms, abelian categories, and complexes in abelian categories. Abstract homology functors are the subject of Chapter 6, with special emphasis on derived functors, Ext and Tor, sheaf cohomology via Godement resolutions, Čech cohomology, and on the Riemann-Roch theorem for compact Riemann surfaces. Chapter 7 analyzes the functors Ext and Tor more closely, including their exact sequences, Baer sums, cotorsion groups, and the universal coefficient theorem for homology in its different versions. Chapter 8 turns to the more advanced homological algebra of rings and modules. The reader meets here the basic dimension theory for rings and modules, Hilbert’s syzygy theorem, stably free modules and Serre’s theorem on finitely generated projective modules over polynomial rings, and the theorem of Auslander-Buchsbaum-Serre on regular local noetherian rings. Chapter 9 gives an ample treatment of the homological aspects of group theory, including group extensions, group cohomology, Schur multipliers, Tate groups, outer automorphisms of finite \(p\)-groups, cohomological dimension of groups, division rings, and Brauer groups. The final Chapter 10 developes the machinery of spectral sequences via exact couples, with many applications to groups, rings, sheaves, and Künneth theorems in (co)homology.

Apart from the many enhancements and improvements, the current new edition comes with an updated bibliography and with an additional index of notation. Also, there is a much larger number of instructive examples and exercises in comparison to the first edition of the book, and the author has added a section titled “How to read this book” for the convenience of the reader. 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 it of new generations of readers.

Reviewer: Werner Kleinert (Berlin)

### MSC:

18-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to category theory |

13-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to commutative algebra |

13Dxx | Homological methods in commutative ring theory |

14Fxx | (Co)homology theory in algebraic geometry |

20Jxx | Connections of group theory with homological algebra and category theory |

18Gxx | Homological algebra in category theory, derived categories and functors |

18-02 | Research exposition (monographs, survey articles) pertaining to category theory |

18G05 | Projectives and injectives (category-theoretic aspects) |

18G10 | Resolutions; derived functors (category-theoretic aspects) |

18G15 | Ext and Tor, generalizations, Künneth formula (category-theoretic aspects) |

18G20 | Homological dimension (category-theoretic aspects) |

18G35 | Chain complexes (category-theoretic aspects), dg categories |

18G40 | Spectral sequences, hypercohomology |

13C10 | Projective and free modules and ideals in commutative rings |

13C11 | Injective and flat modules and ideals in commutative rings |

13C15 | Dimension theory, depth, related commutative rings (catenary, etc.) |

13D05 | Homological dimension and commutative rings |

13D15 | Grothendieck groups, \(K\)-theory and commutative rings |

13D25 | Complexes (MSC2000) |

13E05 | Commutative Noetherian rings and modules |

13E10 | Commutative Artinian rings and modules, finite-dimensional algebras |

13F05 | Dedekind, Prüfer, Krull and Mori rings and their generalizations |

13G05 | Integral domains |

13H05 | Regular local rings |

16Gxx | Representation theory of associative rings and algebras |

16Exx | Homological methods in associative algebras |

16D40 | Free, projective, and flat modules and ideals in associative algebras |

16P60 | Chain conditions on annihilators and summands: Goldie-type conditions |

16D50 | Injective modules, self-injective associative rings |

20J05 | Homological methods in group theory |