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Algebraic K-Theory and Its Applications (Graduate Texts in Mathematics): Jonathan Rosenberg

Algebraic K-Theory plays an important role in many areas of modern mathematics: most notably algebraic topology, number theory, and algebraic geometry, but even including operator theory. The broad range of these topics has tended to give the subject an aura of inapproachability. This book, based on a course at the University of Maryland in the fall of 1990, is intended to enable graduate students or mathematicians working in other areas not only to learn the basics of algebraic K-Theory, but also to get a feel for its many applications. The required prerequisites are only the standard one-year graduate algebra course and the standard introductory graduate course on algebraic and geometric topology. Many topics from algebraic topology, homological algebra, and algebraic number theory are developed as needed. The final chapter gives a concise introduction to cyclic homology and its interrelationship with K-Theory.

Summary: Very effective and understandable overview of K-theory

Rating: 5

Speaking somewhat loosely, algebraic K-theory could be viewed as an attempt to generalize the invariants of linear algebra (such as determinants and canonical forms) to the case of projective modules. In modern mathematical classification, it is to be distinguished from topological K-theory, although both have a lot in common in terms of the general mathematical machinery used and were motivated from similar research interests. This book gives a superb overview of algebraic K-theory, and could be read by anyone who has taken a course in commutative algebra or a course in the theory of rings. The reader will see a common theme throughout algebraic K-theory, namely that of abelianization, which is very prevalent throughout modern mathematics.

In chapter 1, the author begins the construction of K0. After defining projective modules (over a ring R with a unit), and he characterizes finitely generated projective R-modules. The isomorphism classes of finitely generated projective R-modules form an abelian semigroup, the completion of which is the well-known Grothendieck group, and which is defined to be K0(R). The author also develops an alternative approach to K0 using idempotent matrices, thus making a connection with the general linear group. He shows both approaches are essentially equivalent, via the well-known "Morita invariance." Several examples of K0 are discussed, such as when R is a principal ideal domain, a local ring, and a Dedekind domain. The author also discusses to what extent K-theory can be viewed as a "homology for rings". This is relative K-theory, which is defined for two-sided ideals in R. Also, he gives the reader a taste of topological K-theory.

Just as one studies linear transformations of vector spaces and their invariants in linear algebra, the study of automorphisms of free and projective modules is done in K-theory, particularly via the construction of the K1 functor. In chapter 3, the author constructs K1 via the use of matrices, with a more categorical approach delayed until chapter 4. The group of "elementary matrices" for a ring R is defined, and K1(R) is defined to be GL(R)/E(R). The vanishing of K1(R) is equivalent to saying that every matrix in GL(R) and be row or column reduced to the identity matrix. When R is a field, the calculation of K1(R) reduces to ordinary linear algebra via the use of the determinant, and as a consequence K1(R) becomes trivial. When R is a local ring, then there exists a generalization of the determinant, which induces an isomorphism between K1(R) and the matrix group of its units modulo its commutator. When R is a Euclidean ring, K1(R) is isomorphic to its group of units, but when R is a principal ideal domain or a Dedekind domain, the author shows that K1(R) is not so elementary. He again gives a brief discussion of the application of K-theory, via the K1 functor, to topology. The relative K1 theory is also discussed.

Chapter 4 is an overview of how to construct K-theory for categories, instead of just rings. The categories considered are those that are ones wherein it is sensible to speak of an object as being constructed from more elementary objects, and are "abelian" categories, i.e. those that allow such homological results as the Five-Lemma. For a ring R, such categories include those of the category of finitely generated R-modules, the category of R-modules with a finite-type projective resolution, as well as of course the category of finitely generated projective R-modules. After discussing the connection between K0(R), K1(R) and K1 of the Laurent polynomial ring in R (the Bass-Heller-Swan theorem), the author introduces the notion of "negative K-theory", which gives the construction of an exact sequence of an ideal "arbitrarily far to the left", and thus allows the computation of K0(R/I) given information about R and I.

In chapter 5 the author describes the construction of the K2 functor as accomplished by the mathematician John Milnor. This entails a review of the theory of universal central extensions, which the author does in the first section. Following up on the idea that the K-theory of rings measures in some sense the abelian invariants of the non-abelian group GL(R), for a ring R, K2(R) is related to central extensions of E(R) by abelian groups: K2(R) is the kernel of the map of the universal extension of E(R), called the Steinberg group, to E(R). Some examples of the calculation of K2(R) are given, such as for the case where R is a field, wherein K2(R) is generated by the "Steinberg symbols." If R is a finite field, then K2(R) is zero. The lengthy concentration on the case where R is a field is done in order to point out the connection of K2 with number theory. When R is a field, the well-known Brauer group relates K2(R) to finite-dimensional non-abelian division algebras over R, and the author discusses this is fair detail.

The construction of the higher algebraic K-functors, i.e. Ki(R) for i greater than or equal to 3, is done in chapter 5 using the +-construction due to Daniel Quillen. It is here that topological considerations are brought to the forefront, since the Quillen approach is to construct the higher K-functors in terms of the homotopy groups of a particular space, called the classifying space. In particular, the (higher) K-theory of a ring R is defined as the product of the group of units of the classifying space of R and K0(R). The author shows that this definition of K-theory does coincide with that of K0 and K1 done earlier in the book. A brief but interesting survey (with proofs omitted) of the applications of the higher K-groups ends the chapter.

I did not read the last chapter on cyclic homology, and so its review will be omitted.

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