The first edition of the author’s “Algebraic \(K\)-theory” appeared in 1991 (Zbl 0722.19001). After the great standard textbooks and monographs by R. G. Swan, H. Bass, J. Milnor and D. Quillen, which were published between 1968 and 1973, that is just during the first culmination period of this rather young discipline in algebra, the author’s monograph provided the overdue up-dating of this fast-developing subject in a systematic and comprehensive form. Algebraic \(K\)-theory has been a very active area of research, over the past twenty-five years, which revealed a wealth of connections with (and applications to) abstract algebra, algebraic geometry, algebraic topology, and algebraic number theory. Some recent fundamental results in these areas have been proved by \(K\)-theoretic methods, among them the Merkur’ev-Suslin theorem on Brauer groups of fields and the generalization of Quillen’s localization theorem for singular algebraic varieties due to M. Levin, and these results have effected striking applications to the study of the Chow groups of algebraic cycles in smooth or singular algebraic varieties.
Apart from a very thorough, detailed and deep-going presentation of higher algebraic \(K\)-theory, thereby focussing on Quillen’s ingenious original approach from about twenty years ago [cf.: (1) D. Quillen, “Higher algebraic \(K\)-theory. I”, Lect. Notes Math. 341, 85–147 (1973; Zbl 0292.18004); (2) D. Grayson, “Higher algebraic \(K\)-theory. II”, Lect. Notes Math. 551, 217–240 (1976; Zbl 0362.18015)], the author’s text provided a lucid account on precisely these two more recent aspects (and applications) of algebraic \(K\)-theory, namely the Merkur’ev-Suslin theorem and Levine’s generalization of Quillen’s localization theorem. Although presenting a matter of great depth and complexity, on a very advanced level, the author had succeeded in making the text accessible and enjoyable also for nonspecialists, in particular by having added detailed appendices on background material from topology, category theory, and homology theory.
In the second edition under review, the reader will find two major improvements. The author has added another appendix entitled “Results from algebraic geometry”, which contains all the definitions and results needed to work through the chapter on the \(K\)-theory of rings and schemes (Chapter 5) and the most advanced Chapters 8 and 9 on the Merkur’ev-Suslin theorem and the localization for singular algebraic varieties, respectively. Cross-references to this new appendix, throughout the text, make it a lot easier for non-experts in algebraic geometry to master these particular sections. Secondly, the author has rewritten parts of Chapter 8. Based on a later work of Merkur’ev [cf. A. S. Merkur’ev, “\(K_2\) of fields and the Brauer group”, Contemp. Math. 55, 529–546 (1986; Zbl 0608.12028)], he has replaced the fairly complicated treatment of the Merkur’ev-Suslin theorem by a more elementary, now completely self-contained approach.
Finally, this second edition now appears in \(T_EX\) typescript, which makes the book much more user-friendly, too.
Now as before, the author’s book is a very modern and highly advanced text on higher algebraic \(K\)-theory and some of its geometric applications. In regard of the improvements carried out in this second edition, its rôle as a high-level standard text on the subject has even grown. On the other hand, there are now two other recent textbooks at the reader’s disposal, which have appeared in the meantime, and which serve as an excellent source for both flanking introduction and complementary reading to the text under review.
J. Rosenberg’s book “Algebraic \(K\)-theory and its applications” [Grad. Texts Math. 147 (1994; Zbl 0801.19001)] assumes less background of the reader than the author’s text, is of much more introductory character, but covers different (i.e., more topological and less algebro-geometrical) ground. The complementary aspects of additive \(K\)-theory (or “cyclic homology”) can be found in J.-L. Loday’s excellent textbook “Cyclic homology” (1992; Zbl 0780.18009).
Together with these two recent treatises, the (second edition of the) book under review reflects a great deal of the present state of art in algebraic \(K\)-theory and its various applications.