The K-book: An Introduction to Algebraic K-theory

One of the perks of being mathematicians is that as our professional lives unfold, regardless of our mathematical home, we are called to be fellow travelers to all sorts of nearby mathematical lands. If this should meet our individual needs or desires, we can effect a sort of transitive transit: just go from an allied field to a field allied thereto, and so on — ultimately it’s a huge path-connected affair anyway. It’s always possible to go there and back again, as Tolkien might say, wherever “there” might be. In my own life, my number theoretic work has led to a great deal of (marvelous) time spent learning, for example, algebraic geometry (no surprise there), sheaf theory and homological algebra, and most recently different chunks of topology, especially algebraic topology. In each of these somewhat closely allied areas my studies have been idiosyncratic in the sense of being dictated by some sort of number theoretic pursuit, even if broadly understood.

For me, Charles Weibel is something of a known quantity as far as books go: he wrote one of the most influential books in the game, viz., An Introduction to Homological Algebra. It is, in a sense, the natural sequel to the seminal Homological Algebra of Henri Cartan and Samuel Eilenberg, which appeared in 1956 and was absolutely central to the revolution in homological algebra, as part of algebraic geometry, about to go to full steam in the hands of Jean-Pierre Serre (cf. his gorgeous Faisceaux Algébriques Cohérents) and of course Alexandre Grothendieck. This grand movement in mathematics is still with us, as is evidenced, for instance, by the marvelous developments in arithmetic geometry including the victory Andrew Wiles gained over Fermat’s Last Theorem (indeed, over the Shimura-Taniyama-Weil Conjecture).

Weibel’s Homological Algebra goes much farther than Cartan-Eilenberg, of course, since it is much more up-to-date and a lot of homological algebraic things have happened during the second half of the last century. Going at it is a truly fabulous way to get this material under your belt, whether or not you’ve first read Cartan-Eilenberg. (However, a case can be made that every one should read the latter book, just because it is what it is — if you’re going to claim some scholarly familiarity with English literature, you must have studied some Shakespeare seriously. This is similar.)

All these developments in homological algebra naturally led to parallel and connected developments in such areas as algebraic topology and differential geometry, as well as certain parts of theoretical physics — consider, in the latter. the young and fecund theory of D-branes, to pick but one example.

An area that has seen a great deal of such parallel and allied activity is K-theory, which also goes back to Grothendieck’s work in the late 1950s and in its youth drew in such scholars as Michael Atyiah, Daniel Quillan, and Hyman Bass. Now we have before us an introduction to algebraic K-theory (there are other flavors) by none other than Charles Weibel, which truly is “a consummation devoutly to be wished.”

The book, somewhat playfully titled The K-book, starts off in Weibel’s appealing no-nonsense style: “Algebraic K-theory has two components: the classical theory which centers around the Grothendieck group … of a category and uses explicit algebraic presentations and higher algebraic K-theory which requires topological or homological machinery to define.” And he immediately jumps into the deeper waters: “There are three basic versions of the Grothendieck group …” and “… there are four basic constructions for higher algebraic K-theory …” Then he says that “[a]ll these constructions give the same K-theory of a ring but are useful in various distinct settings … [which] fit together as in [a] table that follows,” which is more in the way of an extremely useful Leitfaden indicating in broad terms how one would use the K-theory tool-kit for number theory, algebraic geometry, algebraic topology, and geometric topology. And we are only at the top of the second page of the book’s Preface.

This testifies to the brisk pace of the book, and also its high quality: Weibel presents his important and elegant subject with the authority of an experienced insider, placing stresses where they should be, presenting motivations and characterizations (always succinctly) so as to familiarize the reader with the shape of the subject (its sharps and flats, so to speak) as he learns it by working through the book, and accordingly endowing this same reader with — eventually — the all-important benefit of Fingerspitzengefühl.

The book is subdivided into six beefy chapters, with Grothendieck’s \(K_0\) apearing in Chapter II, with \(K_1\) and \(K_2\) (of a ring) appearing in Chapter III, and with the other middle chapters (IV and V) dealing with higher K-theory and its fundamental theorems. The final Chapter VI deals with the subject of K-theory for fields, a subject of particular importance to number theory.

The last section of the book, § VI.10, is devoted to the K-theory of the ring of integers and includes a particularly beautiful result, modulo Vandiver’s Conjecture, having to do with the Picard group of a certain algebraic extension of \(\mathbb{Z}\) at a sum of two natural l-th roots of unity for an irregular prime l (to wit: the indicated Picard group has no l-torsion): the form of the higher K-group \(K_n(\mathbb{Z})\) is entirely determined for n modulo 8 — Bernoulli numbers enter the game!

Like the other book by Charles Weibel discussed above, The K-book is a wonderful example of solid pedagogy (for the right audience); in particular, it contains a great number of examples, woven beautifully into the narrative, and excellent exercises. Finally, for an evocative account of how Weibel came to write both An Introduction to Homological Algebra (the “other book” just mentioned) and the book under review, as well as Lecture Notes on Motivic Cohomology which he wrote together with Carlo Mazza and Vladimir Voevodsky, see the last section of the Preface.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.