稳定环的相对K,1和相对K,2.docx

摘要
R上的任意两个稳定同构的有限生成投射模均同构当且仅当R上任意阶全矩阵环S均是正则地稳定环(即:对任意正则元a,b∈S, 若Sa-1-Sb=S,则存在c∈S使得a+cb可逆),(R,R) [9]关于半局部环的西群的结果做了一个补充t即举例说明了当R/rad(R)的直和项中含有
F20 F2时,一般地有墨(R)喾U(R)/U(R)’,这里rad(R)是R
的Jacobson根、 F2是二元域、 U(R)是冗的单位群、 u(R)’是U(R)的换位子群;利用Cohn[22】和Silvester[9,10]提出的泛(universal)和拟泛(quasi—universal)的概念,证明了对任意半局部环R及其理想,有Ⅳl(R,J≥呵(R,,)/y(R,I),这里
u(R,I)=U(R)n(1-4-I),V(R,I)是由{(1-t-ab)(1+ha)_1 f n,b∈R且a或b∈I,1+ab∈u(R))生成的U(R)
礁(n,R,I)=ker(K2(n,R)-÷K2(n,R/s)).第四章证明了当环R
及其理想,满足s总(R,,)条件时,弼(m R,』)具有满稳定性;
并且在SSR2(R,I)条件下计算了恐(R,,)和弼(n,R,n这里
SSR2(R,I)条件是指SR2(R,I)条件成立,且对任意o,b∈I有『_ 单位伪正则元c∈R使得1十(a—c)b∈u(R).对交换半局部环 R,Silvester[11】证明了鲍(R)
被推广到弼(n,R,J)上,其中,是半局部环R的理想且包含于月的中心,n≥,通过环
R的Jacobson根的加群结构,给出了当冗是交换有限的局部环、
Jacobson根平方为零且2∈u(R)时的%(R)的结构.
Abstract
This paper considers the lower K—groups under certain stable range Chapter 2,it is proved that any two stable isomorphic finitely generated projective R-moduli are isomorphic ifr any full matrix ring S over R is regularly stable —for any regular elements a,b∈S with Sa+Sb=S,there exists a c∈S such that a+c6∈u(S)).Characterizations of regularly stable rings are also —local rings are important rings satis— lying Bass’s SR2(冗,R)condition Chapter 3 gives plemen—
tarity to Silvester[9]’S result about the KI group of a semi—local
ring,.,an example is given to show that if R/rad(R)contains F2 o F2 as a summand,then KL(R)喾U(R)/U(R)’in general. Using the conception of universal and quasi—universal proposed by Cohn[22】and Silvester[9,101,it is also proved that for any semi—
local ring R and its ideal I,K1(R,I)兰U(R,S)/V(R,n where
U(R,I)=U(R)n(t+I)and V(R,,)is the subgroup ofU(R)gener- atedby{(1+06)(1+ba)一1 I o,b∈Rwith aor 6∈I,l+ab∈u(R)). Let矗j(n,R,I)=ker(K2(n,R)—+.配(n,R/S)).Chapter 4 proves that the弼(n,R,I)group has surjective stability if the SR2(R,I) condition is satisfied,and calcula