The cancellation problem over Noetherian one-dimensional domains

Abstract

Let $R$ be a commutative Noetherian one-dimensional domain containing $\mathbb{Q}$. In this paper we prove that if an $R$-algebra $A$ is such that $A^{\left[n\right]}{\cong }_R{R}^{[n+2]}$, for some $n\ge 1$, then $A{\cong }_R{R}^{\left[2\right]}$. In terms of affine fibrations this means that every stably trivial ${\mathbb{A}}^2$-fibration over $R$ is actually trivial. On the other hand, it is known that this result does not hold in general if $R$ has dimension at least two or if $R$ does not contain $\mathbb{Q}$.