����K��Փ���䑪��Algebraic K-theory and its applications ���ݺ���

����Algebraic K-theory is 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. It associates to any ring R a sequence of abelian groups Ki��R��. The first two of these�� K0 and K1�� are easy to describe in concrete terms�� the others are rather mysterious. For instance�� a finitely generated projective R-module defines an element of K0��R���� and an invertible matrix over R has a "determinant" in K1��R��. The entire sequence of groups K1��R�� behaves something like a homology theory for rings.����Algebraic K-theory plays an important role in many areas�� especially number theory�� algebraic topology�� and algebraic geometry. For instance�� the class group of a number field is essentially K0��R���� where R is the ring of integers�� and "Whitehead torsion" in topology is essentially an element of K1��Z���� where �� is the fundamental group of the space being studied. K-theory in algebraic geometry is basic to Grothendieck's approach to the Riemann-Roch problem. Some formulas in operator theory�� involving determinants and determinant pairings�� are best understood in terms of algebraic K-theory. There is also substantial evidence that the higher K-groups of fields and of rings of integers are related to special values of L-functions and encode deep arithmetic information.