An Algebraic Introduction to K-Theory
This is a fine introduction to algebraic K-theory, requiring only a basic preliminary knowledge of groups, rings and modules. While developing the theory, the author provides numerous examples and presents the basics of closely related areas where K-theory is applied: these include Dedekind domains, classical groups, character theory, quadratic forms, tensor products, completion and localisation, symmetric and tensor algebras, central simple algebras, and Brauer groups of fields.
Part I deals with K0, which creates a kind of substitute for the dimension of arbitrary modules. The Grothendieck group K0(R) of the category of finitely generated projective modules is first introduced as an abelian group, and the related notion of stability is discussed in detail. The ring structure on K0(R) is then introduced via tensor product in the case when R is commutative; as examples, the Witt-Grothendieck ring and the Witt ring are considered in detail and applied to a classification of forms.
Part II deals with applications of K0 to number theory and the representation theory of finite groups. The relation between K0(R) and the class group of a Dedekind domain R is established. Among other things, the structure of semisimple rings and Maschke’s theorem are proved, as well as basic properties of characters of finite groups.
Part III deals with K1(R), which consists of the row-equivalence classes of invertible matrices over R. K1(R) is used to study (generalised) determinants. It is shown that the stable rank of R leads to a bound on the dimensions of the matrices needed to represent K1(R). As an application, the solution of the congruence subgroup problem over Dedekind domains of arithmetic type is presented.
Part IV introduces K2(R) as a fine measure for row reduction of matrices over R. A method of its computation via Steinberg symbols is developed, and the ‘relative exact sequence’ is presented, linking K0, K1 and K2 for R and R/J, where J is an ideal of R. Finally, Matsumoto’s theorem relating K2(R) to symbol maps is proved.
Part V, ‘Sources of K2’ presents various symbol maps occurring in number theory and non-commutative algebra, thereby linking these areas with K-theory. The book culminates with a discussion of the relation between the Brower group of a field F and K2(F), presenting (without proof) the Tate-Merkurjev-Suslin isomorphisms and their consequences for the structure of Azumaya F-algebras. Given the wide range of applications of K-theory presented, the book will certainly be of interest to algebraists, and to number theorists, topologists, geometers and functional analysts.