The first sentence of the Preface of the book under review tells us that relative homological algebra was introduced ca. 1965 by Eilenberg and Moore. It’s fair to say that while every one knows the meaning of the last two words of the three, the same probably cannot be said for the three words put together: what is the meaning of the qualifier? Well, here, verbatim, is the first paragraph of the 1956(!) Trans. AMS paper by Gerhard Hochschild, titled, yes, “Relative homological algebra”:
The main purpose of this paper is to draw attention to certain functors, exactly analogous to the functors ‘Tor’ and ‘Ext’ of Cartan-Eilenberg… , but applicable to a module theory that is relativized with respect to a given subring of the basic ring of operators. In particular, we shall show how certain relative cohomology theories for groups, rings, and Lie algebras can be subsumed under the theory of the relative Ext functor, just as (in [Cartan-Eilenberg]) the ordinary cohomology theories have been subsumed under the theory of the ordinary Ext functor.
This paper by Hochschild appears as item 113 in the bibliography of the book by Enochs and Jenda, so the authors’ decision to identify the actual birth of the subject proper with the 1965 article by Eilenberg-Moore (specifically their AMS Memoir titled Foundations of Relative Homological Algebra) presumably has to do with the apparent fact that the latter presents more of an axiomatic or systematic development of the whole business at hand, while Hochschild’s paper is possibly more along the lines of breaking new ground. After all, Cartan-Eilenberg’s Homological Algebra itself also dates to 1956 (a very good year indeed!).
In any event, homological algebra (à la Cartan-Eilenberg) indeed subsumes (co)homology theories of different flavors under one formalism, namely the yoga of satellites and derived functors of the tensor product functor and of Hom. Tor arises as a left derived functor of the tensor product, while Ext arises as a right derived functor of Hom: see p. 106 of loc.cit. Thus, the topic at hand is really quite familiar after all, at least in large part. In other words, we’re dealing, ab initio, with an old favorite, but then relativization is thrown in, as Hochschild suggests in his comments quoted above. To give a better idea what’s going on, then, and staying with Hochschild for the moment, in his paper’s first section, dealing with relative injective and projective modules, the R-modules that appear are regarded also as S-modules for a fixed subring S of R, and then R-S modules (injective or projective) take center stage, just as in Cartan-Eilenberg we’re dealing simply with R-modules.
Regarding the present book, then, we should note first that an awful lot has happened in the last half-century in the subject of relative homological algebra, as well as homological algebra itself, and this is heavily reflected in what Enochs and Jenda are up to. Regarding the functors singled out above, for example, they note that “the basic functors Hom and Tensor are balanced using resolutions different from the usual projective, injective, and flat resolutions [!] [and this] allows us to compute useful versions of the Extension and Torsion functors with negative indices.” Aha! Already a horse of a different color: this book is several evolutionary steps removed from the sources mentioned above and is really something of a state-of-the-art introduction to the subject.
And so we come to what Enochs and Jenda themselves say about their book: “This book is aimed at graduate students. For that reason, we have attempted to make the book a reasonably self-contained treatment of the subject requiring only familiarity with basic notions in module and ring theory at the level of Basic Algebra I by N. Jacobson.” They go on with a presentation of an outline of the book’s contents. Suffice it to say that the usual suspects appear, as they must: chain conditions, flatness, injective modules, etc. One major feature worth singling out, already hinted at in the preceding paragraph, is Enochs and Jenda’s inclusion of a good deal of material of comparatively recent vintage: “We now have in hand more theorems [than was the case in the early days] guaranteeing the existence of precovers, covers, preenvelopes, and envelopes. These are [nowadays] basic objects of the subject and are used to construct resolutions and then left and right derived functors…” — the nuts and bolts of computations, be that as it may.
Other things covered include cotorsion theories and a lot of material carrying the names of Gorenstein, Iwanaga, and Cohen-Macaulay, ringing bells in the ears of even a number theorist like me for whom homological algebra is primarily a (beloved!) toolkit but not and end in itself.
All this having been said, Enochs and Jenda’s book is clearly a work of high-level scholarship — its present second edition testifies to its success, and rightly so. I count Cartan-Eilenberg as a “desert-island book”; Relative Homological Algebra dovetails beautifully with it.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.