On low-dimensional cancellation problems

A well-known cancellation problem of Zariski asks when, for two given domains (fields) K-1 and K-2 over a field k, a k-isomorphism of K-1[t] (K-1(t)) and K-2[t] (K-2(t)) implies a k-isomorphism of K-1 and K-2. The main results of this article give affirmative answer to the two low-dimensional cases of this problem: 1. Let K be an affine field over an algebraically closed field k of any characteristic. Suppose K(t) similar or equal to k(t(1), t(2), t(3)), then K similar or equal to k(t(1), t(2)). 2. Let M be a 3-diniensional affine algebraic variety over an algebraically closed field k of any characteristic. Let A = K[x, y, z, w]/M be the coordinate ring of M. Suppose A[t] similar or equal to k[x(1), x(2), x(3), x(4)], then frac(A) similar or equal to k (x(1), x(2), x(3)), where frac(A) is the field of fractions of A. In the case of zero characteristic these results were obtained by Kang in [Ming-chang Kang, A note on the birational cancellation problem, J. Pure Appl. Algebra 77 (1992) 141-154; Ming-chang Kang, The cancellation problem, J. Pure Appl. Algebra 47 (1987) 165-171]. However, the case of finite characteristic is first settled in this article, that answered the questions proposed by Kang in [Ming-chang Kang, A note on the birational cancellation problem, J. Pure Appl. Algebra 77 (1992) 141-154; Ming-chang Kang, The cancellation problem, J. Pure Appl. Algebra 47 (1987) 165-171]. (C) 2008 Elsevier Inc. All rights reserved.