´úÊýKÀíÂÛ¼°ÆäÓ¦ÓÃAlgebraic K-theory and its applications

´úÊýKÀíÂÛ¼°ÆäÓ¦ÓÃAlgebraic K-theory and its applications ÄÚÈݼò½é

¡¡¡¡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¦Ð£©£¬ where ¦Ð 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.