The Bass-Quillen problem on a class of local rings with weak global dimension two

Let (R, m) be a local GCD domain. R is called a U ₂ ring if there is an element u ∈ m − m² such that R/(u) is a valuation domain and R _u is a Bézout domain. In this case u is called a normal element of R. In this paper we prove that if R is a U ₂ ring, then R and Rx] are coherent; moreover, if R has a normal element u and dim(R/(u)) = 1, then every finitely generated projective module over RX] is free. This work was supported by the National Natural Science Foundation of China (Grant No. 10671137) and the Ph. D. Programs Foundation of Ministry of Education of China (Grant No. 20060636001)