Eléments de mathématique : algèbre, Chapitre 10, algèbre homologique / N. Bourbaki

At the end of the book, there is the obligatory, immense collection of further-leading exercises, grouped with respect to the different sections of this chapter, as it is notorious for the classical Bourbaki style. Alas, another outstanding feature of the earlier Bourbaki treatises, namely the familiar supplement of respective historical notes, presumably inspired by Bourbaki's former disciplinarian, Jean Dieudonné, is not found in this volume. Nevertheless, the rich collection of about 160 exercises absolutely follows the good old Bourbaki tradition in regard of their comprehensiveness, versality, systematic representation, degree of difficulty, and propelling character. In fact, various topics not explicitely touched upon in the text, as for example regular local rings, spectral sequences, group cohomology, Galois cohomology, Lie algebra cohomology, simplicial schemes, coherent modules, and many other related concepts, are subject to extended exercises, with concrete hints for solution amply provided. On the other hand, many current topics of homological algebra do not occur in the present booklet of 216 pages. First of all, homological algebra is here restricted to the category of modules over a ring, and any categorical or functorial aspects of general homological algebra are consequently left out, including derived functors, adjoint functors, satellites, and other powerful standard constructions of conceptual significance. Well, Bourbaki remained true to their traditional, often criticized conception of not integrating the categorical framework in their systematic representation, which is no particular surprise. On the other hand, there are meanwhile numerous excellent textbooks on general homological algebra for further reading, which comfortably compensate this drawback of Bourbaki's text, and which can be used to solve many of Bourbaki's tough exercises, in a very educating manner. The main feature of Bourbaki's text on homological algebra is its fairly elementary, self-contained and very detailed exposition, besides its traditional methodological compactness. (Zentralblatt)

Notes bibliogr. Index