Cancellation problems and dimension theory

Abstract

A well-known cancellation problem of Zariski asks - for two given domains (fields, respectively) K 1 and K 2 over a field k - whether the k -isomorphism of K1[t] (K(t), respectively) and K2[t] (K2(t), respectively) implies the k -isomorphism of K1 and K2. In this article, we address and systematically study a related problem: whether the k-embedding from K1[t](K1(t), respectively) into K2[t](K2(t), respectively) implies the k-embedding from K 1 into K2. Our results are affirmative: if K1 and K 2 are affine domains over an arbitrary field k, and K1[t] can be k -embedded into K2 [t], then K 1 can be k -embedded into K2; if K1 and K2 are affine fields over an arbitrary field k, and K1 t) can be k-embedded into K2 (t), then K1 can be k-embedded into K2. Similar results are obtained for some general nonaffine domains and nonaffine fields. These results were obtained in Belov et al. (preprint) together with L. Makar-Limanov. In this article we give an alternative proof, show connection with dimension theory, consider the case of infinite transcendental degree, and present some applications and surroundings.