Already in the first paragraph of the book under review, Bruns and Gubeladze unabashedly point out that the juxtaposition of the subject of polytopes with that of rings and K-theory is apt to give one pause: “At first sight, polytopes, by their very nature, must appear alien to surveyors of this heartland of algebra.” But it is in point of fact not unnatural at all to go from polytopes to affine monoids (indeed, one is reminded right off of the baby-steps one takes when first learning algebraic topology: consider the first stage of developing simplicial homology), and lo!, there’s algebraic structure looming on the horizon in the extremely suggestive form of toric varieties. It’s subsequently almost inevitable that one has to travel on in the direction of K-theory.
Polytopes, Rings, and K-Theory weighs in at over 400 pages and ten chapters split into three main parts, culminating in the aforementioned K-theory in the given context. The prevailing approach is classical (modulo the modernity of K-theory as such — but this characterization may just be an indication of my full-fledged middle-age) and as constructible as is reasonable: witness the appearance of Gröbner bases in Chapter 7. There are exercises galore in the book, “ranging from mere tests on the basic notions to guided tours of research… [replete with] extensive hints or pointers to the literature.”
Bruns and Gubeladze have attempted to make the geometry (for lack of a better word) self-contained: “The theory of polyhedra and affine monoids is developed from scratch;” on the other hand, the algebra requires more preliminary preparation from the reader, for which the authors recommend Atiyah and Macdonald (unquestionably the best book from which to learn commutative algebra “the first time around,” if I may be excused a bit of editorializing) and Lam’s Serre’s Problem on Projective Modules (which was reviewed in this column some time ago, if I may be excused a bit of self-promotion).
All in all, Polytopes, Rings, and K-Theory is an accessible and well-written book on an interesting and important subject (and the presence of toric varieties in the game underscores this point). It should be quite a success.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.