##
**Homology.**
*(German)*
Zbl 0133.26502

Die Grundlehren der mathematischen Wissenschaften. Bd. 114. Berlin-Göttingen-Heidelberg: Springer-Verlag. x, 522 pp. with 7 fig. (1963).

Homological algebra was extracted from its surrounding mathematical milieu by H. Cartan and S. Eilenberg [Homological Algebra. Princeton: Princeton University Press (1956; Zbl 0075.24305)] who identified it as an object for study and use by topologists and algebraists, and described and analysed its state at the time of writing. Since then the subject has matured – to a considerable extent due to the impetus they provided, but also due to its own internal capacity for development – and the time was surely ripe for the appearance of the present volume which introduces the reader to virtually all the main aspects of the theory. But it is not by any means simply its timeliness which commends this book. It is an extremely well-organized text and thoroughly readable. Topics are presented in such a way that the reader is made aware of the source of the topic, of its principal features and of its applicability. In particular many notions appear first in rather special contexts (e.g., the bar construction). This policy enables the reader to get a firm grip on the idea before he is led – easily and gently – into the appropriate generalization. He is then led once again back from the general to the particular so that the new mathematical concept becomes meaningful and significant for him. It is more a tribute to their insight than a criticism of their style of presentation to say that the authors of the earlier text provided their readers with no such ready access to the acquisition of genuine understanding; for the applications of homological algebra have expanded prodigiously since 1956 and for many of the ideas extracted by Cartan-Eilenberg from the surrounding mathematical context only the original source constituted at the time an active domain for the application of the idea.

However, one feature of the present work which distinguishes it from all previous texts devoted to homological algebra and which particularly appeals to this reviewer is the frequent explicit recognition of topology as the source of the algebraical ideas being described. The author is ready to describe the topological background of the really important notions of homological algebra (e.g., the cohomology of groups); and he also shows how developments within homological algebra themselves feed back into topology and provide important information and stimulate the evocation of new topological concepts. The author is not at all concerned to refine homological algebra to the extent that it be utterly uncontaminated by alien notions from topology! On the contrary, the very title of the book seems to the reviewer to indicate the author’s liberal attitude to his subject.

There are further important features of this book. The cross-referencing and referencing are excellent (the author has been remarkably conscientious in providing an up-to-date bibliography) and the reader is told just where he can find treatments of further developments of particular topics. There are historical and ancillary notes appended to many sections and chapters to clarify sources and to introduce alternative treatments or supplementary material. There are exercises at the end of each section which are (almost always) thoroughly cognate to the material and, moreover, in some cases lead the reader into interesting new developments. Set against this array of fine features, the reviewer’s complaints are very minor. It is strange, for example, that in a book on homology it is so difficult to locate the definition of \(\mathrm{Ext}^n\) (the index is useless for this purpose!); similarly, Opext does not occur in the index. Hopf algebras are talked about and exemplified with no mention of the homology or cohomology of \(H\)-spaces and Hopf’s classical paper is omitted from the bibliography. There is an unexplained but clearly deliberate irruption of a \(\pm\) sign on p. 117 (Corollary 5.4).

The reviewer would have liked, as a matter of personal opinion, to have seen room made for an introductory treatment of Amitsur cohomology, and perhaps a few other topics. But these mild criticisms and opinions do not mitigate the reviewer’s opinion that this is a splendid addition to the literature, of a, quality worthy of one who has contributed so much to the development and under-standing of its subject matter.

There follows a brief summary of the contents of the book. Following a discursive Introduction, Chapter I, Modules, Diagrams and Functors, is of a preparatory nature.

Chapter II, Homology of Complexes, views homology theory from the algebraic and topological standpoint, including a description of singular homology and the statement of its homotopy invariance.

In Chapter III, Extensions and Resolutions, the extension problem for modules serves to motivate the definition of \(\mathrm{Ext}\) and hence, by the Yoneda generalization, the definition of \(\mathrm{Ext}^n\). Projective and injective resolutions are introduced and \(\mathrm{Ext}^n\) is characterized in terms of these resolutions. The connecting homomorphism is first met in the very concrete form of a homomorphism \(\operatorname{Hom}(A,G) \to\mathrm{Ext}(C, G)\) associated with a short exact sequence \(A\to B\to C\) of \(R\)-modules, a good example of the author’s style of exposition to which we have already referred.

Chapter IV, Cohomology of Groups, is introduced by the study of group extensions and further motivated by copious reference to the topological origins of the theory. Building up from a discussion of \(H^1\) and \(H^2\), the author is led naturally to introduce the bar construction.

In Chapter V, Tensor and Torsion Products, the author motivates his definition of Torn by means of the explicit description of \(\mathrm{Tor}_n\) for abelian groups, by generators and relations; the equivalence with the definition by means of projective resolutions is then immediately provided. The Künneth formula is established from the author’s definition.

Chapter VI, Types of Algebras, discusses various algebras and coalgebras over a ground ring, and contains a section on the identities relating \(\otimes\) and \(\operatorname{Hom}\).

Chapter VII, Dimension, contains a great wealth of “classical” material on homological dimension with applications to polynomial rings (Koszul complex, Hilbert’s Chain of Syzygies theorem), separable algebras, and local rings.

Chapter VIII, Products, gives an extensive listing of the products encountered in the theory, including the coproduct (diagonal) in a category of simplicial objects; and a proof of the Eilenberg-Zilber theorem is provided via acyclic models.

Chapter IX is essentially introductory to Chapter X; though entitled Relative Homological Algebra, its first three chapters are concerned with very general category-theoretical preliminaries, including the definition of an abelian category. Then the notions of an allowable class of short exact sequences and allowable projective (injective) objects lead to that of a relative abelian category (surprisingly, not in the list of categories in the index, but in the list of abelian categories) and so to relative homological algebra. But the principal applications are postponed to Chapter X, Cohomology of Algebraic Systems, where the bar resolution for algebras is defined and leads to the description of the Hochschild cohomology. The homology of algebras, groups and modules is also discussed in this chapter. The bar construction is applied to the homology and cohomology theory of DGA-algebras and commutative DGA-algebras, but the work of Cartan in the latter connection, though referred to, is not discussed. The final section of this chapter is a survey of the literature and the current (at the time of writing) state of affairs in the general area of homology of algebraic systems; this remains an extremely valuable compendium of information for the reader.

Chapter XI, on Spectral Sequences, is an admirable piece of exposition. After the necessary algebraic ideas have been given, the second section is devoted to describing without proof their application (Leray-Serre) to fiber spaces and to showing specific results which may be obtained (e.g., the homology of the loop space on a sphere). This interplay of algebraic theory and motivation (through topology, group theory or whatever) continues throughout the chapter.

Chapter XII, Derived Functors, generalizes the context of many of the results and concepts treated in earlier chapters. It has, avowedly, a far more abstract flavor than its predecessors, but will doubtless be regarded as thoroughly concrete by algebraists of the “new wave”. It treats such topics as diagram chasing and the embedding theorem for abelian categories, \(\mathrm{Ext}\) without projectives, connected sequences of functors, definitions by universal properties, and, of course, derived functors. These topics are all motivated by applications already familiar to the reader. In fact, the author brings together material from Chapter XI and XII most effectively in the final section in deriving the spectral Künneth Theorem.

However, one feature of the present work which distinguishes it from all previous texts devoted to homological algebra and which particularly appeals to this reviewer is the frequent explicit recognition of topology as the source of the algebraical ideas being described. The author is ready to describe the topological background of the really important notions of homological algebra (e.g., the cohomology of groups); and he also shows how developments within homological algebra themselves feed back into topology and provide important information and stimulate the evocation of new topological concepts. The author is not at all concerned to refine homological algebra to the extent that it be utterly uncontaminated by alien notions from topology! On the contrary, the very title of the book seems to the reviewer to indicate the author’s liberal attitude to his subject.

There are further important features of this book. The cross-referencing and referencing are excellent (the author has been remarkably conscientious in providing an up-to-date bibliography) and the reader is told just where he can find treatments of further developments of particular topics. There are historical and ancillary notes appended to many sections and chapters to clarify sources and to introduce alternative treatments or supplementary material. There are exercises at the end of each section which are (almost always) thoroughly cognate to the material and, moreover, in some cases lead the reader into interesting new developments. Set against this array of fine features, the reviewer’s complaints are very minor. It is strange, for example, that in a book on homology it is so difficult to locate the definition of \(\mathrm{Ext}^n\) (the index is useless for this purpose!); similarly, Opext does not occur in the index. Hopf algebras are talked about and exemplified with no mention of the homology or cohomology of \(H\)-spaces and Hopf’s classical paper is omitted from the bibliography. There is an unexplained but clearly deliberate irruption of a \(\pm\) sign on p. 117 (Corollary 5.4).

The reviewer would have liked, as a matter of personal opinion, to have seen room made for an introductory treatment of Amitsur cohomology, and perhaps a few other topics. But these mild criticisms and opinions do not mitigate the reviewer’s opinion that this is a splendid addition to the literature, of a, quality worthy of one who has contributed so much to the development and under-standing of its subject matter.

There follows a brief summary of the contents of the book. Following a discursive Introduction, Chapter I, Modules, Diagrams and Functors, is of a preparatory nature.

Chapter II, Homology of Complexes, views homology theory from the algebraic and topological standpoint, including a description of singular homology and the statement of its homotopy invariance.

In Chapter III, Extensions and Resolutions, the extension problem for modules serves to motivate the definition of \(\mathrm{Ext}\) and hence, by the Yoneda generalization, the definition of \(\mathrm{Ext}^n\). Projective and injective resolutions are introduced and \(\mathrm{Ext}^n\) is characterized in terms of these resolutions. The connecting homomorphism is first met in the very concrete form of a homomorphism \(\operatorname{Hom}(A,G) \to\mathrm{Ext}(C, G)\) associated with a short exact sequence \(A\to B\to C\) of \(R\)-modules, a good example of the author’s style of exposition to which we have already referred.

Chapter IV, Cohomology of Groups, is introduced by the study of group extensions and further motivated by copious reference to the topological origins of the theory. Building up from a discussion of \(H^1\) and \(H^2\), the author is led naturally to introduce the bar construction.

In Chapter V, Tensor and Torsion Products, the author motivates his definition of Torn by means of the explicit description of \(\mathrm{Tor}_n\) for abelian groups, by generators and relations; the equivalence with the definition by means of projective resolutions is then immediately provided. The Künneth formula is established from the author’s definition.

Chapter VI, Types of Algebras, discusses various algebras and coalgebras over a ground ring, and contains a section on the identities relating \(\otimes\) and \(\operatorname{Hom}\).

Chapter VII, Dimension, contains a great wealth of “classical” material on homological dimension with applications to polynomial rings (Koszul complex, Hilbert’s Chain of Syzygies theorem), separable algebras, and local rings.

Chapter VIII, Products, gives an extensive listing of the products encountered in the theory, including the coproduct (diagonal) in a category of simplicial objects; and a proof of the Eilenberg-Zilber theorem is provided via acyclic models.

Chapter IX is essentially introductory to Chapter X; though entitled Relative Homological Algebra, its first three chapters are concerned with very general category-theoretical preliminaries, including the definition of an abelian category. Then the notions of an allowable class of short exact sequences and allowable projective (injective) objects lead to that of a relative abelian category (surprisingly, not in the list of categories in the index, but in the list of abelian categories) and so to relative homological algebra. But the principal applications are postponed to Chapter X, Cohomology of Algebraic Systems, where the bar resolution for algebras is defined and leads to the description of the Hochschild cohomology. The homology of algebras, groups and modules is also discussed in this chapter. The bar construction is applied to the homology and cohomology theory of DGA-algebras and commutative DGA-algebras, but the work of Cartan in the latter connection, though referred to, is not discussed. The final section of this chapter is a survey of the literature and the current (at the time of writing) state of affairs in the general area of homology of algebraic systems; this remains an extremely valuable compendium of information for the reader.

Chapter XI, on Spectral Sequences, is an admirable piece of exposition. After the necessary algebraic ideas have been given, the second section is devoted to describing without proof their application (Leray-Serre) to fiber spaces and to showing specific results which may be obtained (e.g., the homology of the loop space on a sphere). This interplay of algebraic theory and motivation (through topology, group theory or whatever) continues throughout the chapter.

Chapter XII, Derived Functors, generalizes the context of many of the results and concepts treated in earlier chapters. It has, avowedly, a far more abstract flavor than its predecessors, but will doubtless be regarded as thoroughly concrete by algebraists of the “new wave”. It treats such topics as diagram chasing and the embedding theorem for abelian categories, \(\mathrm{Ext}\) without projectives, connected sequences of functors, definitions by universal properties, and, of course, derived functors. These topics are all motivated by applications already familiar to the reader. In fact, the author brings together material from Chapter XI and XII most effectively in the final section in deriving the spectral Künneth Theorem.

Reviewer: P. J. Hilton

### MSC:

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

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

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

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 |