Cancellation for two-dimensional unique factorization domains

We show that if A and B are finitely generated two-dimensional unique factorization domains over an algebraically closed field, then A vertical bar x vertical bar congruent to B vertical bar x vertical bar implies A congruent to B. The proof is an application of an algebraic technique involving the AK invariant which has previously been used to obtain other cancellation theorems. (C) 2009 Elsevier B.V. All rights reserved.