Algebraic K-theory and its applications.

*(English)*Zbl 0801.19001
Graduate Texts in Mathematics. 147. New York, NY: Springer-Verlag. x, 392 p. (1994).

A very welcome and accessible introduction to algebraic \(K\)-theory with an enormous amount of motivational remarks, exercises and also sidetracks to topological \(K\)-theory, number theory, algebraic geometry, operator theory and functional analysis.

In the preface algebraic \(K\)-theory is presented as ‘the branch of algebra dealing with linear algebra (especially in the limiting case of large matrices) over a general ring \(R\) instead of over a field’. On reading this book one will soon be convinced of the tremendous wealth of this linear algebra over general rings. At the same time, because of the author’s very stimulating style and choice of material, the reader may risk to become an assiduous student of \(K\)-theory.

The book consists of a preface, six chapters, references, a notational index and a subject index. The setup of the first two and the fourth chapters follows grosso modo J. Milnor’s famous “Introduction to algebraic \(K\)-theory” [Ann. Math. Stud. 72 (Princeton, 1971; Zbl 0237.18005)]. The \(K_ 0\), \(K_ 1\) and \(K_ 2\) of rings are extensively studied with special emphasis on local rings, principal ideal domains and Dedekind domains, where many concrete computations, many in the form of step by step exercises, give a nice idea of the strength of algebraic \(K\)-theory. Also, for a ring \(R\) with ideal \(I\subset R\), the relative \(K\)-groups \(K_ 0(R,I)\) and \(K_ 1(R,I)\) are introduced and the exact sequence relating \(K_ 0\), \(K_ 1\) and \(K_ 2\) of the ring \(R\), the relative \(K_ i(R,I)\) (\(i=0,1\)) and the \(K_ 0\), \(K_ 1\) and \(K_ 2\) of the quotient \(R/I\) is constructed in several steps throughout the various chapters. In the third chapter, i.e. before the introduction of Milnor’s \(K_ 2\) in chapter four, \(K_ 0\) and \(K_ 1\) (as well as \(G_ 0\) and \(G_ 1\)) of categories with exact sequences and small skeleton are introduced, and, of course, it is shown that for a ring \(R\), \(K_ i(\text{Proj } R)\) (\(i=0,1\)), where \(\text{Proj } R\) is the category of finitely generated projective \(R\)-modules, may be identified in a natural way with \(K_ i(R)\) as defined in chapters one and two. As closely related subjects, computations and applications of \(K_ 0\), \(K_ 1\) and \(K_ 2\) are treated in detail with necessary prerequisites on topology, algebraic geometry, Hilbert-Schmidt and trace-class operators, homological algebra etc. explained on the way, we just mention Swan’s theorem on the relation between algebraic and topological \(K_ 0\) (f.g. projective modules vs. sections of vector bundles over compact Hausdorff spaces), Euler characteristics and Wall’s finiteness obstruction in algebraic topology, Whitehead groups and Whitehead torsion, Mennicke symbols, the Grothendieck/Bass-Heller-Swan theorems which assert that for a left Noetherian ring \(R\) one has natural isomorphisms \(G_ 0(R){\overset \sim {} }G_ 0(R [t,t^{-1}])\) and correspondingly for the \(K_ 0\)’s in case \(R\) is also regular, similarly (not quite the same) isomorphisms for the \(G_ 1\)’s, and \(K_ 1\)’s of number fields, in particular, \(K_ 2(\mathbb{Q})\), Mercurjev-Suslin (without proof), second Whitehead group in the topology of manifolds, \(\dots\), etc.

The fifth chapter deals with classifying spaces and Quillen’s \(+\)- construction. Here it is shown how to regard the \(K\)-groups \(K_ i(R)\) (\(i=0,1,2\)) and \(K_ i(R,I)\) (\(i=0,1\)) as homotopy groups \(\pi_ i(\mathbb{K}(R))\) and \(\pi_ i(\mathbb{K}(R,I))\) of suitable CW-complexes with basepoint \(\mathbb{K}(R)\) and \(\mathbb{K}(R,I)\). As a matter of fact, \(\mathbb{K}(R,I)\) is just the homotopy fiber of the map \(q^*\colon \mathbb{K}(R)\to q^* (\mathbb{K}(R)) \subset \mathbb{K} (R/I)\) induced by the quotient map \(q\colon R\to R/I\). This makes it possible to define higher \(K\)-groups \(K_ i(R)\) and \(K_ i(R,I)\) and extends the above mentioned exact sequence. The chapter ends with a survey of some twenty pages on higher \(K\)-theory. Here the proofs are rather sketchy, but one gets an idea of the kind of mathematics involved. Subjects treated are e.g. Quillen’s results on the \(K\)-groups of finite fields and the algebraic closure \(\overline {\mathbb{F}}_ p\), \(K\)-theory with finite coefficients and related results of Browder and of Suslin, \(K_ 3(\mathbb{Z})\), Quillen’s and Borel’s results on the higher \(K\)-groups of rings of integers of number fields, Lichtenbaum’s result on the rank of the higher \(K\)-groups \(K_{2k+1}(R)\) of the ring of integers \(R\) of the number field \(F\) and the order of the zero of the zeta-function \(\zeta_ F(s)\) at \(s=- k\). Finally, a brief survey of Quillen’s \(Q\)-construction and some of its applications is given.

The last chapter gives an outline of cyclic homology and its relation to \(K\)-theory. Cyclic homology is introduced as a modification of Hochschild homology. Connes’ long exact sequence is discussed as well as the connections with non-commutative de Rham theory. A detailed account of the construction of the Chern character for higher \(K\)-groups is presented. The book ends with a variety of applications of cyclic homology and the Chern character formalism.

Altogether one may recommend to anyone interested in learning from scratch ‘\(K\)-theory and its applications’ (and much more) to read and enjoy Rosenberg’s enthusing book.

In the preface algebraic \(K\)-theory is presented as ‘the branch of algebra dealing with linear algebra (especially in the limiting case of large matrices) over a general ring \(R\) instead of over a field’. On reading this book one will soon be convinced of the tremendous wealth of this linear algebra over general rings. At the same time, because of the author’s very stimulating style and choice of material, the reader may risk to become an assiduous student of \(K\)-theory.

The book consists of a preface, six chapters, references, a notational index and a subject index. The setup of the first two and the fourth chapters follows grosso modo J. Milnor’s famous “Introduction to algebraic \(K\)-theory” [Ann. Math. Stud. 72 (Princeton, 1971; Zbl 0237.18005)]. The \(K_ 0\), \(K_ 1\) and \(K_ 2\) of rings are extensively studied with special emphasis on local rings, principal ideal domains and Dedekind domains, where many concrete computations, many in the form of step by step exercises, give a nice idea of the strength of algebraic \(K\)-theory. Also, for a ring \(R\) with ideal \(I\subset R\), the relative \(K\)-groups \(K_ 0(R,I)\) and \(K_ 1(R,I)\) are introduced and the exact sequence relating \(K_ 0\), \(K_ 1\) and \(K_ 2\) of the ring \(R\), the relative \(K_ i(R,I)\) (\(i=0,1\)) and the \(K_ 0\), \(K_ 1\) and \(K_ 2\) of the quotient \(R/I\) is constructed in several steps throughout the various chapters. In the third chapter, i.e. before the introduction of Milnor’s \(K_ 2\) in chapter four, \(K_ 0\) and \(K_ 1\) (as well as \(G_ 0\) and \(G_ 1\)) of categories with exact sequences and small skeleton are introduced, and, of course, it is shown that for a ring \(R\), \(K_ i(\text{Proj } R)\) (\(i=0,1\)), where \(\text{Proj } R\) is the category of finitely generated projective \(R\)-modules, may be identified in a natural way with \(K_ i(R)\) as defined in chapters one and two. As closely related subjects, computations and applications of \(K_ 0\), \(K_ 1\) and \(K_ 2\) are treated in detail with necessary prerequisites on topology, algebraic geometry, Hilbert-Schmidt and trace-class operators, homological algebra etc. explained on the way, we just mention Swan’s theorem on the relation between algebraic and topological \(K_ 0\) (f.g. projective modules vs. sections of vector bundles over compact Hausdorff spaces), Euler characteristics and Wall’s finiteness obstruction in algebraic topology, Whitehead groups and Whitehead torsion, Mennicke symbols, the Grothendieck/Bass-Heller-Swan theorems which assert that for a left Noetherian ring \(R\) one has natural isomorphisms \(G_ 0(R){\overset \sim {} }G_ 0(R [t,t^{-1}])\) and correspondingly for the \(K_ 0\)’s in case \(R\) is also regular, similarly (not quite the same) isomorphisms for the \(G_ 1\)’s, and \(K_ 1\)’s of number fields, in particular, \(K_ 2(\mathbb{Q})\), Mercurjev-Suslin (without proof), second Whitehead group in the topology of manifolds, \(\dots\), etc.

The fifth chapter deals with classifying spaces and Quillen’s \(+\)- construction. Here it is shown how to regard the \(K\)-groups \(K_ i(R)\) (\(i=0,1,2\)) and \(K_ i(R,I)\) (\(i=0,1\)) as homotopy groups \(\pi_ i(\mathbb{K}(R))\) and \(\pi_ i(\mathbb{K}(R,I))\) of suitable CW-complexes with basepoint \(\mathbb{K}(R)\) and \(\mathbb{K}(R,I)\). As a matter of fact, \(\mathbb{K}(R,I)\) is just the homotopy fiber of the map \(q^*\colon \mathbb{K}(R)\to q^* (\mathbb{K}(R)) \subset \mathbb{K} (R/I)\) induced by the quotient map \(q\colon R\to R/I\). This makes it possible to define higher \(K\)-groups \(K_ i(R)\) and \(K_ i(R,I)\) and extends the above mentioned exact sequence. The chapter ends with a survey of some twenty pages on higher \(K\)-theory. Here the proofs are rather sketchy, but one gets an idea of the kind of mathematics involved. Subjects treated are e.g. Quillen’s results on the \(K\)-groups of finite fields and the algebraic closure \(\overline {\mathbb{F}}_ p\), \(K\)-theory with finite coefficients and related results of Browder and of Suslin, \(K_ 3(\mathbb{Z})\), Quillen’s and Borel’s results on the higher \(K\)-groups of rings of integers of number fields, Lichtenbaum’s result on the rank of the higher \(K\)-groups \(K_{2k+1}(R)\) of the ring of integers \(R\) of the number field \(F\) and the order of the zero of the zeta-function \(\zeta_ F(s)\) at \(s=- k\). Finally, a brief survey of Quillen’s \(Q\)-construction and some of its applications is given.

The last chapter gives an outline of cyclic homology and its relation to \(K\)-theory. Cyclic homology is introduced as a modification of Hochschild homology. Connes’ long exact sequence is discussed as well as the connections with non-commutative de Rham theory. A detailed account of the construction of the Chern character for higher \(K\)-groups is presented. The book ends with a variety of applications of cyclic homology and the Chern character formalism.

Altogether one may recommend to anyone interested in learning from scratch ‘\(K\)-theory and its applications’ (and much more) to read and enjoy Rosenberg’s enthusing book.

Reviewer: W.W.J.Hulsbergen (Haarlem)

##### MSC:

19-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to \(K\)-theory |

19-02 | Research exposition (monographs, survey articles) pertaining to \(K\)-theory |