赋值环上的Suslin稳定性定理

The Suslin's stability theorem on valuation rings

本文证明了赋值环上的Suslin稳定性定理,研究并得到:当n3时,任意赋值环V上的特殊线性群SL_(n)(V[x])可以由该环上初等矩阵群E_(n)(V[x])生成,即SL_(n)(V[x])中每一个矩阵都可以分解成初等矩阵的乘积.进一步证明了,对于任意算术环R,当n3时,SL_(n)(R[x])=SL_(n)(R)·E_(n)(R[x]).

In this paper,we prove that the Suslin's stability theorem holds for an arbitrary valuation ring V,and the special linear group SL_n(V[x])coincides with the group generated by elementary matrices En(V[x])for n 3,i.e.,every matrix in SLn(V[x])can be factorized as a product of elementary matrices.For any arithmetical ring R and n 3,we show that SL_n(R[x])=SL_n(R)·E_n(R[x]).