On low-dimensional cancellation problems

Abstract

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 ) ≃ k ( t 1 , t 2 , t 3 ) , then K ≃ k ( t 1 , t 2 ) . 2. Let M be a 3 -dimensional 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 ] ≃ k [ x 1 , x 2 , x 3 , x 4 ] , then frac ( A ) ≃ 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].