Applications of Algebraic $K$-Theory to Algebraic Geometry and Number Theory

This volume presents a state-of-the-art description of some of the exciting applications of algebraic $K$-theory to other branches of mathematics, especially algebraic geometry and algebraic number theory. As the proceedings of a 1983 AMS-IMS-SIAM Joint Summer Research Conference, it includes current and important work by some of the best researchers in the field. The diverse scope includes the following topics: the matrix/vector bundle tradition of concrete computations for specific rings, the interaction with algebraic cycles, and the generalization of the regulator map for units in an algebraic number field to higher $K$-groups of varieties over number fields. Of particularly high research value are the ideas of Beilinsonon, which are presented here for the first time, the work of Merkurjev and Suslin relating $K$-theory to the Brauer group (as reported by Merkurjev and Wadsworth), and the papers by Kato on algebraic cycles. Directed towards mathematicians working in algebraic $K$-theory, algebraic geometry, and algebraic number theory, this volume is also of interest to the algebraic topologist. The reader should be familiar with basic $K$-theory and interested in its applications to other areas of mathematics.