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Title: Introduction to Some Methods of Algebraic K-Theory (Regional conference series in mathematics) by Hyman Bass ISBN: 0-8218-1670-5 Publisher: Amer Mathematical Society Pub. Date: 01 June, 1982 Format: Paperback List Price(USD): $24.00

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Average Customer Rating: 4 (1 review)

Rating: 4
Summary: Short but adequate
Comment: Written for those who are not familiar to the subject, the author gives a brief but fairly effective overview of the subject of algebraic K-theory. Once thought to be of interest only to pure mathematicians, K-theory has now found applications to high energy physics and cryptography. Higher algebraic K-theory is not covered, the author realizing that very advanced technical machinery is needed for such material. The notes could be used though as an introduction to higher K-theory and Quillen's construction of K-groups for categories with exact sequences.

The author begins the lectures with stating the main goal of the book, namely for proving that for the polynomial ring A in N variables over the integers or integers modulo p, the general linear group of n by n matrices GL(n,A) over this ring is finitely generated for n greater than or equal to N + 3. To meet this goal he reviews the properties of elementary matrices in lecture 1. For a ring A, by considering the elementary subgroups E(n,A) of GL(n,A), these subgroups consisting of matrices satisfying certain relations, the author shows that for a surjective ring homomorphism between rings A and A', the homomorphism from E(n,A) to E(n,A') is surjective, even though it is not for GL(n,A) to GL(n,A'). E(n,A) is shown to be stable under transposition and shown to be commutator subgroup of GL(n,A) for large n. This is the origin of the stability issues in K-theory, and these are discussed in lecture 2. The author shows just why it is advantageous to consider taking the union GL(A) of GL(n,A) (and E(n,A)) for all n and why stability is important in the proof of the above result.

The "Whitehead group" K(1,A) is defined as GL(A)/E(A), and its use in the proof of the theorem results from the map GL(n,A)/E(n,A) to K(1,A) being a bijection for large n and that K(1,A) is finitely generated. Following this matrix characterization of K(1,A), the author reduces the proof of the theorem to showing that for a "regular" ring A, every unipotent element in GL(A) represents 0 in K(1,A), and that the rings in the theorem are indeed regular. Noting the analogy between determinants of matrices and determinants of endomorphisms of vector spaces, the author begins the proof of these assertions with a different description of K(1,A). This description involves the consideration of Grothendieck and Whitehead groups of categories with exact sequences.

The Whitehead group is now defined as the quotient of the Grothendieck group, the latter being the abelian group whose generators are essentially isomorphism classes of objects from an admissible Abelian category. The Whitehead group K(1,A) for a ring A is then related to the Whitehead group K(1,M) for an admissible category M. This definition is due to Grothendieck and involves showing that their is an isomorphism between K(1,A) and K(1, P(A)) where P(A) is the category of finitely generated projective A-modules. P(A) is not abelian, and therefore must be enlarged, without changing K(1,A), to one that is. The author shows that P(A) must be abelian in order to kill unipotents K(1,A). The enlarged P(A) is abelian as long as A is regular, the latter meaning that A is right Noetherian and that any finitely generated A-module has finite homological dimension. As the name implies, homological dimensions involves some discussion of homology theory, and is defined to the least n for which there is a projective resolution of the A-module of length n. The proof of the above theorem then follows, as the author shows, from Hilbert's syzygy theorem.