In Threading Homology Through Algebra: Selected Patterns, Giandomenico Boffi and David Buchsbaum present a somewhat different contribution to the intersection of algebra and topology, i.e. to algebraic topology, here with the emphasis placed emphatically on the adjective. Specifically, the authors characterize their work with the following description: “The expectation is that an educated reader will see connections between areas that he had not seen before, and will learn techniques that will help [him] in further research in these areas.” So the book really is something of an idiosyncratic labor of love — love for a particularly attractive part of algebra (modulo the proviso that one is ab initio kindly disposed to what some uncharitably have called “abstract nonsense” — does this phrase really go back to Eilenberg himself? And is it just me, or is this description heard less and less these days?).
In the Preface, Boffi and Buchsbaum go on to state that their goal is to show how Koszul complexes and their variations, and resolutions in general, “affect the perception of certain problems in selected parts of algebra, as well as their success in solving a number of them.” By definition (cf. p. 39 of the book under review), if f : A → R is an R-module homomorphism, let K(f)p be the p-th exterior power of A, and define the differential df by the natural rule df(a1 ∧ … ∧ ap) = ∑ (-1)if(ai)a1∧ … ∧ âi∧ … ∧ ap; get quickly that df2 = 0 and df(a ∧ b ) = df(a ) ∧ b + (-1)pa ∧ df (b ). Then the Koszul complex K(f) is obtained from these K(f)p, with p ∈ N, in the usual way. Koszul complexes manifestly occur rather frequently in nature, particularly in local ring theory. And Boffi and Buchsbaum quickly expand their charter to include generalized Koszul complexes and various resolutions, e.g. finite free ones, to which a full chapter is devoted.
Qua applications, so to speak, the latter parts of Threading Homology Through Algebra concern such things as exactness criteria and Weyl and Schur modules, underscoring the fact that novices should steer clear of the book, notwithstanding a very nice preparatory section (of about forty pages) dealing with linear and multilinear algebra. Again, let the authors speak for themselves: “The intended readership of this book ranges from third-year and above graduate students in mathematics, to the accomplished mathematician who may or may not be in any of the fields touched on, but would like to see what developments have taken place in these areas and perhaps launch himself into some of the open problems suggested.” Fair enough.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles.