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The \(K\)-book. An introduction to algebraic \(K\)-theory. (English) Zbl 1273.19001

Graduate Studies in Mathematics 145. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-9132-2/hbk). xii, 618 p. (2013).
\(K\)-theory as an independent discipline in pure mathematics emerged about fifty years ago. From its very beginnings, \(K\)-theory was divided into algebraic and topological \(K\)-theory, according to its various conceptual contents and applications. Roughly speaking, algebraic \(K\)-theory deals with functorial invariants for rings, fields, schemes and varieties, algebraic sheaves and vector bundles, or more general categories with particular structure, and its role in modern algebraic geometry, algebraic number theory, algebraic topology as well as in the theory of operator algebras within functional analysis is nowadays a ubiquitous, utmost crucial one. Among the classical research monographs and textbooks in algebraic \(K\)-theory, most of which were published during the last thirty years of the 20th century, there are the well-known works of H. Bass [Algebraic K-theory. W. A. Benjamin, Inc., New York-Amsterdam (1968; Zbl 0174.30302)], R. Swan [ Lect. Notes Math. 76 (1968; Zbl 0193.34601)], J. W. Milnor [Ann. Math. Stud. No. 72 (1971; Zbl 0237.18005)], V. Srinivas [Progress in Mathematics (Boston, Mass.) 90. Boston, MA: Birkhäuser. xvi, 341 p. (1996; Zbl 0860.19001)], J. Rosenberg [Algebraic \(K\)-theory and its applications. Graduate Texts in Math. 147. New York, NY: Springer-Verlag (1994; Zbl 0801.19001)], H. Inassaridze [Algebraic K-theory. Math. Appl. 311 (1995; Zbl 0836.19001)] and B. A. Magurn [An algebraic introduction to \(K\)-theory. Cambridge: Cambridge University Press (2002; Zbl 1002.19001)]. Each of these classics reflects its own spirit of the age and comes with its own particular focus, thereby imparting its respective individual viewpoint of the rapidly developing subject of algebraic \(K\)-theory.
The book under review aims to provide another comprehensive introduction to the principles of algebraic \(K\)-theory, with the special intention to combine the classical approaches with more recent topological techniques for higher algebraic \(K\)-theory, on the one hand, and to describe some of the topical applications of the latter in contemporary algebraic geometry and number theory, on the other hand. In this regard, the present textbook of algebraic \(K\)-theory takes the reader from the classical basics of the subject to the present state of the art, leading her/him in such a way to the forefront of current research in the field.
As the author, one of the leading experts in algebraic \(K\)-theory, points out in the preface to his new “\(K\)-book” in hand, this book project of his grew steadily since the mid-1990s, when the \(K\)-theory landscape had significantly changed, and when all the venerable classics in the field (see above) had appeared. Actually, the growing of this book could be followed up on the author’s web page at Rutgers University, USA, ever since, and the complete book can still be downloaded from there under the address: http:/www.math.rutgers.edu/~weibel/.
As for the contents, the book consists of six chapters, each of which is subdivided into several sections. Chapter 1 gives an introduction to the basic objects studied in algebraic \(K\)-theory: projective modules over a ring and vector bundles over algebraic schemes. This includes such standard material as free modules and stably free modules, projective modules, the Picard group of a commutative ring, topological vector bundles and their Chern classes as well as the very basic facts on sheaves of modules and algebraic vector bundles over schemes in algebraic geometry. Here the presentation is rather survey-like, with many instructive concrete examples and explanations, whereas hints to proofs of crucial results are given in the exercises at the end of this introductory chapter. Several ways to construct the Grothendieck group \(K_0\) of a mathematical object are described in Chapter 2 ranging from the group completion of a monoid to \(K_0\) of an exact category.
Along the way, the reader meets the related groups \(K(X)\), \(K0(X)\), and \(KU(X)\) of a topological space \(X\), lambda-rings and their Adams operations, the \(K_0\)-group of schemes and varieties, before a section on Waldhausen categories and their \(K_0\)-theory concludes this chapter.
The classical constructions of the functors \(K_1\) and \(K_2\) are the topics discussed in Chapter 3, where most of the material is presented in the form of a brief overview.
Apart from the Whitehead group \(K_1\) of a ring and its relative version, the fundamental theorems relating \(K_1\) and \(K_2\), H. Bass’s concept of \(K_{-1}\) and \(K_{-2}\) and an overview of J. Milnor’s theory of the group \(K_2(\mathbb{R})\) of a ring \(\mathbb{R}\), also Steinberg groups, Steinberg symbols, and Matsumoto’s theorem on \(K_2(F)\) of a field \(F\), also an analogue of Hilbert’s Theorem 90 for \(K_2\)-groups of cyclic field extensions are touched upon. This chapter ends with a further-going discussion of Milnor’s general groups \(K^M_n(F)\) associated to a field \(F\).
Chapter 4 is devoted to the four standard constructions for higher groups: Quillen’s \(\text{BGL}^+\)-construction for rings via homotopy groups of certain topological spaces, the group completion constructions for symmetric monoidal categories via the homotopy theory of topological \(H\)-spaces, Quillen’s \(Q\)-construction for exact categories, and the so-called “Waldhausen \(wS\)-construction” for Waldhausen categories. Again, a wealth of topics, in this context, is taken up, but only very few proofs of theorems are presented, as the focus is on a panoramic, explanatory presentation of the advanced, highly abstract and sophisticated material. The subsequent Chapter 5 presents the fundamental structure theorems for the constructions in higher \(K\)-theory as outlined in Chapter 4 all of which give the same \(K\)-theory in the special case of a base ring. The author restricts the attention to exact categories and Waldhausen categories, using the extra structure of those to derive the extensions to higher \(K\)-theory of the various structure theorems for \(K_0\) discussed in Chapter 2. Among the numerous applications of higher \(K\)-theory in algebra and algebraic geometry are several cases of the so-called Gersten conjecture on discrete valuation domains and the interpretation of the \(K\)-cohomology of regular quasi-projective schemes in terms of Gersten’s coniveau spectral sequence for the higher \(K\)-theory of rings. Finally, Chern classes for rings and schemes are depicted in the context of higher \(K\)-theory.
The concluding Chapter 6 turns to the problem of computing the higher \(K\)-groups of fields. More precisely, the goal of this chapter is to explain what the present state of knowledge of the algebraic \(K\)-theory of (number) fields is, thereby largely illuminating the historical developments from the early 1970s until now. This includes topics such as the \(K\)-theory of algebraically closed fields, the \(K\)-theory of \(\mathbb{R}\), relations to motivic cohomology, \(K_3\) of a field, and various \(K\)-theoretic results for special number fields, local fields, and the ring \(\mathbb{Z}\) of integers.
Each section of the book ends with a large set of related exercises, which are mainly of purely theoretical nature. These exercises mostly refer to additional concepts and theorems from the respective research literature, therefore requiring intensive further reading. However, ample hints to the original papers are given throughout the text, thereby referring to the sweeping bibliography with more than 230 references at the end of the book.
Indeed, Charles Weibel’s “\(K\)-book” offers a plethora of material from both classical and more recent algebraic \(K\)-theory. It is a perfect source book for seasoned graduate students and working researchers, who are willing and eager to follow the author’s expository path, and who are ready for a lot of additional reading and self-reliant work. The many instructive examples and clarifying remarks help the reader grasp the essentials of algebraic \(K\)-theory from a panoramic view, and the entire exposition represents a highly valuable and useful guide to the subject in all its diversity and topicality. Although barely being a textbook for neophytes in the field, despite the wealth of background material sketched wherever necessary, the book under review is certainly the most topical presentation of algebraic \(K\)-theory at this time, and an excellent enhancement of the existing literature in any case.

MSC:

19-00 General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to \(K\)-theory
19-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to \(K\)-theory
19Axx Grothendieck groups and \(K_0\)
19Bxx Whitehead groups and \(K_1\)
19Cxx Steinberg groups and \(K_2\)
19Dxx Higher algebraic \(K\)-theory
19Fxx \(K\)-theory in number theory
19L64 Geometric applications of topological \(K\)-theory
19E08 \(K\)-theory of schemes
19F27 Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects)
19M05 Miscellaneous applications of \(K\)-theory
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Online Encyclopedia of Integer Sequences:

a(n) = order of 2-primary subgroup of the group K_n(Z).