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Relative homological algebra. (English) Zbl 0952.13001
de Gruyter Expositions in Mathematics. 30. Berlin: Walter de Gruyter. xi, 339 p. (2000).
This is a textbook on a rather special topic in the field of homological algebra of rings and modules. The subject of relative homological algebra was introduced by S. Eilenberg and J. C. Moore in their memoir “Foundations of relative homological algebra” [Mem. Am. Math. Soc. 55 (1965; Zbl 0129.01101)]. The basic objects of this framework were precovers, covers, preenvelopes and envelopes of modules, and those were used to construct various kinds of resolutions and their resulting derived functors in module theory. After the appearance of the pioneering work of Eilenberg and Moore, thirty-five years ago, much progress has been made in this field, and the authors of the book under review have contributed a great deal to the development in relative homological algebra.
Now they present another systematic treatment of this subject, including the various new developments and applications. As their book is primarily aimed at graduate students in homological algebra, the authors have made any effort to keep the text reasonably self-contained and detailed. The outcome is a comprehensive textbook on relative homological algebra at its present state of art. The considerably rich material is presented by twelve chapters entitled as follows:
1.
“Basic concepts” (Zorn’s lemma, modules, categories and functors, complexes of modules and homology, direct and inverse limits, completions);
2.
“Flat modules, chain conditions, and prime ideals” (flat modules, localization, chain conditions, primary decomposition, the Artin-Rees lemma, Zariski rings);
3.
“Injective and flat modules” (injective modules, flatness, Matlis duality);
4.
“Torsion-free covering modules” (existence of torsion-free precovers and covers, behavior under direct sums ad products, examples);
5.
“Covers” (precovers and covers with respect to a given class of modules, existence theorems, projective and flat covers, injective covers, behavior under direct sums, nilpotency);
6.
“Envelopes” (preenvelopes and envelopes with respect to a given class of modules, existence theorems, flat envelopes, injective envelopes, pure injective envelopes);
7.
“Covers, envelopes, and cotorsion theories” (fibrations, cofibrations, Wakamatsu lemmas, set-theoretic homological algebra, cotorsion theories with enough injectives and projectives);
8.
“Relative homological algebra and balance” (covers and envelopes in abelian categories, applications to categories of modules, dimension theory);
9.
“Iwanaga-Gorenstein and Cohen-Macaulay rings and their modules” (Iwanaga-Gorenstein rings, minimal injective resolutions, torsion products of injective modules, local cohomology and the dualizing module);
10.
“Gorenstein modules” (Gorenstein injective and projective modules, Gorenstein flat modules, Foxby classes);
11.
“Gorenstein covers and envelopes” (injective precovers, covers, preenvelopes and envelopes for Gorenstein-related classes of modules, Auslander’s theorem on Gorenstein projective covers, Gorenstein flat covers, Gorenstein flat and projective preenvelopes);
12.
“Balance over Gorenstein and Cohen-Macaulay rings” (applications of the general framework of relative homological algebra to the dimension theory of modules over \(n\)-Gorenstein rings and Cohen-Macaulay rings, \(\Omega\)-Gorenstein modules over Cohen-Macaulay rings).
The bibliography is (apparently) absolutely complete and up-to-date. The authors have listed up nearly two hundred references related to the text, and there is a set of bibliographical notes at the end of the book, which provide some detailed hints with respect to both used sources (for the different chapters) and possible further reading. – The exposition of the material is characterized by a righ degree of profundity, comprehensiveness, rigor and clarity. – Each section comes with its own equipment of exercises which frequently lead the reader to refined results.
Apart from being an introduction to, and a reference book for contemporary relative homological algebra, this text is also suitable for an introductory course in commutative and ordinary homological algebra, at least in consideration of the basic material presented in chapters 1, 2, 3 and 9.

MSC:
13-02 Research exposition (monographs, survey articles) pertaining to commutative algebra
18-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to category theory
13D03 (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.)
16-02 Research exposition (monographs, survey articles) pertaining to associative rings and algebras
13C15 Dimension theory, depth, related commutative rings (catenary, etc.)
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
16D40 Free, projective, and flat modules and ideals in associative algebras
16D50 Injective modules, self-injective associative rings
16E10 Homological dimension in associative algebras
16P40 Noetherian rings and modules (associative rings and algebras)
18G10 Resolutions; derived functors (category-theoretic aspects)
18G25 Relative homological algebra, projective classes (category-theoretic aspects)
18G15 Ext and Tor, generalizations, Künneth formula (category-theoretic aspects)
13C14 Cohen-Macaulay modules
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