Cancellation for two-dimensional unique factorization domains (Q1020952)

This surprisingly short paper (approx. one page of preliminaries, one for the proof), a proof is given of cancellation for finitely generated two-dimensional UFDs over an algebraically closed field. To be precise: Let \(A,B\) be finitely generated two-dimensional UFDs over an algebraically closed field. If \(A[x]\cong B[x]\), then \(A\cong B\). The main technique for the proof is the AK-invariant (often denoted ML, as Makar-Limanov invariant). The main difficulty is in avoiding issues with characteristic \(p\), which is done in the introduction, by not considering locally nilpotent derivations, but exponential maps (or in fact ``locally finite iterative higher derivation'' which coincides with ``locally nilpotent'' if the characteristic is zero). Note that a generalisation to the theorem is not possible, due to counterexamples of \textit{D. Finston} and the reviewer [Isr. J. Math. 163, 369--381 (2008; Zbl 1139.14045)]. Also, the assumption UFD cannot be dropped: the famous Danielewski surfaces are counterexamples.