##
**An introduction to homological algebra.**
*(English)*
Zbl 0797.18001

Cambridge Studies in Advanced Mathematics. 38. Cambridge: Cambridge University Press. xiv, 450 p. (1994).

Homological algebra is a mathematical framework that initially emerged from classical simplicial topology, essentially as a crystallization product of the concepts of homology and cohomology groups of topological spaces. Since the early 1940’s, when S. Eilenberg, S. MacLane, and others discovered that the formalism of (co-)homology in topology could be axiomatized, and conveyed to more general algebraic systems, the development of abstract “homological” methods has undergone a rapid expansion, touching almost every area of algebra, geometry and topology.

The first systematic account on homological algebra (as a now self- reliant, mature algebraic theory) was provided by the epoch-making textbook of H. Cartan and S. Eilenberg, published in 1956, which substantially influenced the further development of the subject itself as well as its increasingly far-reaching applications in other branches of mathematics. As homological methods gained in significance, applications and popularity, other textbooks successively appeared on the subject, partially reflecting the further progress made in the meantime, on the one hand, and casually trying to make the topic more accessible (on a comparatively elementary level) to non-specialists, on the other hand.

Among the most frequently used ones, apart from the pioneering work of Cartan and Eilenberg [cf. H. Cartan and S. Eilenberg, Homological Algebra. Princeton University Press (1956; Zbl 0075.24305)], are the well-known texts by D. Northcott [An introduction to homological algebra. Cambridge, U.K.: At the University Press (1960; Zbl 0116.01401)], S. MacLane [Homology. Grundlehren der Mathematischen Wissenschaften, Band 114. Berlin etc.: Springer-Verlag (1963; Zbl 0133.26502)], J. Jans [Rings and homology. New York etc.: Holt, Rinehart & Winston (1964; Zbl 0141.02901)], P. Hilton and U. Stammbach [A course in homological algebra, Berlin: Springer-Verlag (1971; Zbl 0238.18006)], J. Rotman [An introduction to homological algebra. New York: Academic Press (1979; Zbl 0441.18018)], and Bourbaki’s supplementary volume to “Algèbre” [cf. N. Bourbaki, Algèbre homologique, Chapter X of “Algèbre”. Paris: Masson (1980; Zbl 0455.18010)].

During that period, from 1955 to the early 1980’s, the development of homological algebra and its applications proceeded likewise tempestously. Largely due to the influence of A. Grothendieck, J.-P. Serre, and their French school of algebraic geometry, homological algebra has been increasingly formulated in the general terms of Abelian categories and derived functors. This provided the necessary framework for the rigorous foundation of modern algebraic geometry by the theory of schemes and their morphisms, sheaves and their cohomology, local cohomology, and other central notions. Related areas, such as commutative algebra, algebraic topology, algebraic number theory, and algebraic analysis were just as influenced, for instance by homological concepts like spectral sequences, group cohomology, Galois cohomology, Hochschild homology, André-Quillen homology, cyclic homology, triangulated categories, derived categories, \({\mathcal D}\)-modules in microanalysis, etc. Thus the landscape of homological algebra has evolved tremendously since the appearance of the most standard texts.

In these days, homological algebra is a fundamental tool, of obviously still increasing importance, for mathematicians and (nowadays also) theoretical physicists. It represents a basic framework for relating local and global properties of algebraic, geometric, or topological objects, proving non-constructive existence theorems, investigating obstructions to carrying out various kinds of constructions, and serves as a well-established common language in most fields of pure mathematics. Unfortunately, many of the later developments in homological algebra are, of course, not contained in the earlier standard texts and, what is more, also not easily found in the vast original literature. This meanwhile cumbersome situation certainly requires a new text that provides a comprehensive and unified account of homological algebra at its present state of art, and which helps to break down the increasing barrier between active researchers and casual users.

The present book is a rewarding attempt to take this requirement into account. The contents are arranged in such a way that the first half of the book covers the canonical (or classical) topics in homological algebra, as they are also found in most of the earlier standard texts, whereas the second half is devoted to the more recent developments mentioned above. Concretely, the text is subdivided into ten chapters consisting of several sections each, and an appendix providing some basic material from category theory.

Chapter I gives an introduction to chain complexes and the basic operations on them, including a generalization to complexes in Abelian categories.

Chapter II introduces derived functors and delta-functors, following Grothendieck’s original approach via projective and injective resolutions. Along this line, the concept of sheaf cohomology for topological spaces is briefly sketched as an application. This chapter concludes with adjoint functors, Tor and Ext, and exactness properties of functors. The discussion of balancing the functors Ext and Tor is particularly nice and uses a new, elegant approach.

Chapter III deals with the standard material on Ext and Tor. In addition, the author discusses the derived functor \(\lim^ 1\) of the inverse limit of modules, the Künneth formulas, the universal coefficient theorems, and their applications to algebraic topology.

Chapter IV is devoted to the homology theory of rings, including homological dimension theory, rings of small dimension, local rings, Koszul complexes, and the basics of local cohomology “à la Grothendieck”.

Chapter V provides an introduction to spectral sequences. In contrast to the other textbooks, where spectral sequences are usually introduced at the end of the book, and therefore actually never applied, they are here discussed early enough to be able to utilize them as a fundamental tool in the following chapters.

As to the algebraic aspects of spectral sequences, most of the material can be already found in the book of Cartan and Eilenberg. However, the author simultaneously spends much effort at illustrating their computational power, too, mainly by discussing some of the most important topological applications (e.g., the significance of the Leray-Serre spectral sequence in algebraic topology). Another feature is provided by the fact that, for the first time, several results on the convergence properties of spectral sequences are covered, which are difficult to find somewhere else but widely used by topologists.

After that, homology and cohomology of groups is discussed in Chapter VI. The material covered here is basically the canonical one, yet the presentation is exceptionally elegant and unifying. In particular, there are extra sections on the Lyndon/Hochschild-Serre spectral sequence (as a special case of the Grothendieck spectral sequence), covering spaces and classifying spaces in topology, universal central extensions of groups, and Galois cohomology (including profinite groups and Brauer groups).

Another illustration of the homological methods developed so far is given in Chapter VII which explains Lie algebra (co-)homology. The first few sections provide a fairly self-contained introduction to Lie algebras, modules over Lie algebras and universal enveloping algebras, whereas the following sections deal with the classical cohomology theory of Lie algebras initiated in 1948 by C. Chevalley and S. Eilenberg, with special emphasis on universal central extensions and affine Lie algebras. The latter topics are essentially taken from the more recent work of R. Wilson [Euclidean Lie algebras are universal central extensions, Lect. Notes Math. 933, 210–213 (1982; Zbl 0498.17012)].

The following chapters turn to modern developments in homological algebra. Chapter VIII concerns simplicial methods, and that both from the classical and from the modern point of view. This includes discussions on simplicial objects in categories, simplicial homotopy groups, the Dold-Kan correspondence, the Eilenberg-Zilber theorem, canonical resolutions, cotriple homology resolutions, and a short outline of the recent André-Quillen (co-)homology theory in “homotopical algebra”.

Chapter IX gives an introduction to Hochschild (co-)homology and cyclic homology, together with some of their applications. The material on Hochschild (co-)homology and cohomology of algebras is partly classical, and already contained in S. MacLane’s book “Homology” (cited above), on the other hand enriched by more recent relationships to derivations, algebraic differentials, and smooth algebras, following Grothendieck’s approach in “EGA IV” [A. Grothendieck and J. Dieudonné, Éléments de géométrie algébrique, Part IV, Publ. Math., Inst. Hautes Etud. Sci. 20 (1964; Zbl 0136.15901), ibid. 24 (1965; Zbl 0135.39701), ibid. 28 (1966; Zbl 0144.19904), ibid. 32 (1967; Zbl 0153.22301)].

The discussion of cyclic homology is rather new in the textbook literature, and essentially based on the original papers by A. Connes [Cohomologie cyclique et foncteurs \(\text{Ext}^ n\), C. R. Acad. Sci. Paris, Sér. I 296, 953-958 (1983; Zbl 0534.18009)], J.-L. Loday and D. Quillen [Cyclic homology and the Lie algebra homology of matrices, Comment. Math. Helv. 59, 565–591 (1984; Zbl 0565.17006)], and T. Goodwillie [Cyclic homology, derivations, and the free loop space, Topology 24, 187–215 (1985; Zbl 0569.16021)]. Comprehensive textbooks on this subject are just appearing, e.g., J.-L. Loday’s book “Cyclic homology” [Berlin: Springer-Verlag (1992; Zbl 0780.18009)] and the forthcoming book by D. Husemoller and C. Kassel (“Cyclic homology”, book in preparation).

The concluding Chapter X is devoted to the concept of the derived category of an Abelian category. This encompasses the algebraic aspects, as they have been developed by J.-L. Verdier in “SGA \(4{1\over 2}\)” [Lect. Notes Math. 569, 262–311 (1977; Zbl 0407.18008)] and R. Hartshorne [Residues and duality, Lect. Notes Math. 20 (1966; Zbl 0212.26101)], as well as the topological counterpart in stable homotopy theory [L. G. Lewis, J. P. May and M. Steinberger, Equivariant stable homotopy theory, Lect. Notes Math. 1213 (1986; Zbl 0611.55001)]. Another good reference for derived categories, not cited in the present book, is the recent (Russian) text “Methods of homological algebra, Vol. I” by S. I. Gel’fand and Yu. I. Manin [Nauka, Moskva (1988; Zbl 0668.18001)]. Their approach to homological algebra is entirely centered on the notion of derived category, from the very beginning on, and therefore completely different from any other text on homological algebra, including the present one.

Altogether, and as already mentioned above, the present textbook is a highly welcome addition to the existing standard literature on homological algebra, particularly valuable with regard to the well-arranged, methodically unified compound of both the indispensable traditional techniques and some of the most important recent parts of the modern homological toolkit. The many intermingled special topics from various areas in mathematics, among them the historically distinguished topological aspects and applications, endow this text with an extraordinary width and versatility. As prerequisites, only the basic knowledge from an introductory graduate algebra course is assumed; all the notions from higher category theory utilized in this book are summarized in the appendix. Numerous examples and exercises scattered in the text furnish the reader with a plenty of further motivations and hints.

Thus the present book on homological algebra represents both an introductory, self-contained text suitable for graduate students and a very useful, up-to-date reference book for working mathematicians (and physicists) using homological methods.

The first systematic account on homological algebra (as a now self- reliant, mature algebraic theory) was provided by the epoch-making textbook of H. Cartan and S. Eilenberg, published in 1956, which substantially influenced the further development of the subject itself as well as its increasingly far-reaching applications in other branches of mathematics. As homological methods gained in significance, applications and popularity, other textbooks successively appeared on the subject, partially reflecting the further progress made in the meantime, on the one hand, and casually trying to make the topic more accessible (on a comparatively elementary level) to non-specialists, on the other hand.

Among the most frequently used ones, apart from the pioneering work of Cartan and Eilenberg [cf. H. Cartan and S. Eilenberg, Homological Algebra. Princeton University Press (1956; Zbl 0075.24305)], are the well-known texts by D. Northcott [An introduction to homological algebra. Cambridge, U.K.: At the University Press (1960; Zbl 0116.01401)], S. MacLane [Homology. Grundlehren der Mathematischen Wissenschaften, Band 114. Berlin etc.: Springer-Verlag (1963; Zbl 0133.26502)], J. Jans [Rings and homology. New York etc.: Holt, Rinehart & Winston (1964; Zbl 0141.02901)], P. Hilton and U. Stammbach [A course in homological algebra, Berlin: Springer-Verlag (1971; Zbl 0238.18006)], J. Rotman [An introduction to homological algebra. New York: Academic Press (1979; Zbl 0441.18018)], and Bourbaki’s supplementary volume to “Algèbre” [cf. N. Bourbaki, Algèbre homologique, Chapter X of “Algèbre”. Paris: Masson (1980; Zbl 0455.18010)].

During that period, from 1955 to the early 1980’s, the development of homological algebra and its applications proceeded likewise tempestously. Largely due to the influence of A. Grothendieck, J.-P. Serre, and their French school of algebraic geometry, homological algebra has been increasingly formulated in the general terms of Abelian categories and derived functors. This provided the necessary framework for the rigorous foundation of modern algebraic geometry by the theory of schemes and their morphisms, sheaves and their cohomology, local cohomology, and other central notions. Related areas, such as commutative algebra, algebraic topology, algebraic number theory, and algebraic analysis were just as influenced, for instance by homological concepts like spectral sequences, group cohomology, Galois cohomology, Hochschild homology, André-Quillen homology, cyclic homology, triangulated categories, derived categories, \({\mathcal D}\)-modules in microanalysis, etc. Thus the landscape of homological algebra has evolved tremendously since the appearance of the most standard texts.

In these days, homological algebra is a fundamental tool, of obviously still increasing importance, for mathematicians and (nowadays also) theoretical physicists. It represents a basic framework for relating local and global properties of algebraic, geometric, or topological objects, proving non-constructive existence theorems, investigating obstructions to carrying out various kinds of constructions, and serves as a well-established common language in most fields of pure mathematics. Unfortunately, many of the later developments in homological algebra are, of course, not contained in the earlier standard texts and, what is more, also not easily found in the vast original literature. This meanwhile cumbersome situation certainly requires a new text that provides a comprehensive and unified account of homological algebra at its present state of art, and which helps to break down the increasing barrier between active researchers and casual users.

The present book is a rewarding attempt to take this requirement into account. The contents are arranged in such a way that the first half of the book covers the canonical (or classical) topics in homological algebra, as they are also found in most of the earlier standard texts, whereas the second half is devoted to the more recent developments mentioned above. Concretely, the text is subdivided into ten chapters consisting of several sections each, and an appendix providing some basic material from category theory.

Chapter I gives an introduction to chain complexes and the basic operations on them, including a generalization to complexes in Abelian categories.

Chapter II introduces derived functors and delta-functors, following Grothendieck’s original approach via projective and injective resolutions. Along this line, the concept of sheaf cohomology for topological spaces is briefly sketched as an application. This chapter concludes with adjoint functors, Tor and Ext, and exactness properties of functors. The discussion of balancing the functors Ext and Tor is particularly nice and uses a new, elegant approach.

Chapter III deals with the standard material on Ext and Tor. In addition, the author discusses the derived functor \(\lim^ 1\) of the inverse limit of modules, the Künneth formulas, the universal coefficient theorems, and their applications to algebraic topology.

Chapter IV is devoted to the homology theory of rings, including homological dimension theory, rings of small dimension, local rings, Koszul complexes, and the basics of local cohomology “à la Grothendieck”.

Chapter V provides an introduction to spectral sequences. In contrast to the other textbooks, where spectral sequences are usually introduced at the end of the book, and therefore actually never applied, they are here discussed early enough to be able to utilize them as a fundamental tool in the following chapters.

As to the algebraic aspects of spectral sequences, most of the material can be already found in the book of Cartan and Eilenberg. However, the author simultaneously spends much effort at illustrating their computational power, too, mainly by discussing some of the most important topological applications (e.g., the significance of the Leray-Serre spectral sequence in algebraic topology). Another feature is provided by the fact that, for the first time, several results on the convergence properties of spectral sequences are covered, which are difficult to find somewhere else but widely used by topologists.

After that, homology and cohomology of groups is discussed in Chapter VI. The material covered here is basically the canonical one, yet the presentation is exceptionally elegant and unifying. In particular, there are extra sections on the Lyndon/Hochschild-Serre spectral sequence (as a special case of the Grothendieck spectral sequence), covering spaces and classifying spaces in topology, universal central extensions of groups, and Galois cohomology (including profinite groups and Brauer groups).

Another illustration of the homological methods developed so far is given in Chapter VII which explains Lie algebra (co-)homology. The first few sections provide a fairly self-contained introduction to Lie algebras, modules over Lie algebras and universal enveloping algebras, whereas the following sections deal with the classical cohomology theory of Lie algebras initiated in 1948 by C. Chevalley and S. Eilenberg, with special emphasis on universal central extensions and affine Lie algebras. The latter topics are essentially taken from the more recent work of R. Wilson [Euclidean Lie algebras are universal central extensions, Lect. Notes Math. 933, 210–213 (1982; Zbl 0498.17012)].

The following chapters turn to modern developments in homological algebra. Chapter VIII concerns simplicial methods, and that both from the classical and from the modern point of view. This includes discussions on simplicial objects in categories, simplicial homotopy groups, the Dold-Kan correspondence, the Eilenberg-Zilber theorem, canonical resolutions, cotriple homology resolutions, and a short outline of the recent André-Quillen (co-)homology theory in “homotopical algebra”.

Chapter IX gives an introduction to Hochschild (co-)homology and cyclic homology, together with some of their applications. The material on Hochschild (co-)homology and cohomology of algebras is partly classical, and already contained in S. MacLane’s book “Homology” (cited above), on the other hand enriched by more recent relationships to derivations, algebraic differentials, and smooth algebras, following Grothendieck’s approach in “EGA IV” [A. Grothendieck and J. Dieudonné, Éléments de géométrie algébrique, Part IV, Publ. Math., Inst. Hautes Etud. Sci. 20 (1964; Zbl 0136.15901), ibid. 24 (1965; Zbl 0135.39701), ibid. 28 (1966; Zbl 0144.19904), ibid. 32 (1967; Zbl 0153.22301)].

The discussion of cyclic homology is rather new in the textbook literature, and essentially based on the original papers by A. Connes [Cohomologie cyclique et foncteurs \(\text{Ext}^ n\), C. R. Acad. Sci. Paris, Sér. I 296, 953-958 (1983; Zbl 0534.18009)], J.-L. Loday and D. Quillen [Cyclic homology and the Lie algebra homology of matrices, Comment. Math. Helv. 59, 565–591 (1984; Zbl 0565.17006)], and T. Goodwillie [Cyclic homology, derivations, and the free loop space, Topology 24, 187–215 (1985; Zbl 0569.16021)]. Comprehensive textbooks on this subject are just appearing, e.g., J.-L. Loday’s book “Cyclic homology” [Berlin: Springer-Verlag (1992; Zbl 0780.18009)] and the forthcoming book by D. Husemoller and C. Kassel (“Cyclic homology”, book in preparation).

The concluding Chapter X is devoted to the concept of the derived category of an Abelian category. This encompasses the algebraic aspects, as they have been developed by J.-L. Verdier in “SGA \(4{1\over 2}\)” [Lect. Notes Math. 569, 262–311 (1977; Zbl 0407.18008)] and R. Hartshorne [Residues and duality, Lect. Notes Math. 20 (1966; Zbl 0212.26101)], as well as the topological counterpart in stable homotopy theory [L. G. Lewis, J. P. May and M. Steinberger, Equivariant stable homotopy theory, Lect. Notes Math. 1213 (1986; Zbl 0611.55001)]. Another good reference for derived categories, not cited in the present book, is the recent (Russian) text “Methods of homological algebra, Vol. I” by S. I. Gel’fand and Yu. I. Manin [Nauka, Moskva (1988; Zbl 0668.18001)]. Their approach to homological algebra is entirely centered on the notion of derived category, from the very beginning on, and therefore completely different from any other text on homological algebra, including the present one.

Altogether, and as already mentioned above, the present textbook is a highly welcome addition to the existing standard literature on homological algebra, particularly valuable with regard to the well-arranged, methodically unified compound of both the indispensable traditional techniques and some of the most important recent parts of the modern homological toolkit. The many intermingled special topics from various areas in mathematics, among them the historically distinguished topological aspects and applications, endow this text with an extraordinary width and versatility. As prerequisites, only the basic knowledge from an introductory graduate algebra course is assumed; all the notions from higher category theory utilized in this book are summarized in the appendix. Numerous examples and exercises scattered in the text furnish the reader with a plenty of further motivations and hints.

Thus the present book on homological algebra represents both an introductory, self-contained text suitable for graduate students and a very useful, up-to-date reference book for working mathematicians (and physicists) using homological methods.

Reviewer: W. Kleinert (Berlin)

### MSC:

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

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

16Exx | Homological methods in associative algebras |

18E30 | Derived categories, triangulated categories (MSC2010) |

20F99 | Special aspects of infinite or finite groups |

17B55 | Homological methods in Lie (super)algebras |

55Uxx | Applied homological algebra and category theory in algebraic topology |