Noncommutative analogues of a cancellation theorem of Abhyankar, Eakin, and Heinzer

摘要

Letkbe a field and letAbe a finitely generatedk-algebra. The algebraAis said to be cancellative if wheneverBis anotherk-algebra with the property that A[x] congruent to B[x] then we necessarily have A congruent to B. An important result of Abhyankar, Eakin, and Heinzer shows that ifAis a finitely generated commutative integral domain of Krull dimension one then it is cancellative. We consider the question of cancellation for finitely generated not-necessarily-commutative domains of Gelfand-Kirillov dimension one, and show that such algebras are necessarily cancellative when the characteristic of the base field is zero. In particular, this recovers the cancellation result of Abhyankar, Eakin, and Heinzer in characteristic zero when one restricts to the commutative case. We also provide examples that show affine domains of Gelfand-Kirillov dimension one need not be cancellative when the base field has positive characteristic, giving a counterexample to a conjecture of Tang, the fourth-named author, and Zhang. In addition, we prove a skew analogue of the result of Abhyankar-Eakin-Heinzer, in which one works with skew polynomial extensions as opposed to ordinary polynomial rings.