On the triviality of an ${\mathbb{A}}^2$-fibration over a DVR

Abstract

In this paper we show that any ${\mathbb{A}}^2$-fibration over a discrete valuation ring which is also an ${\mathbb{A}}^2$-form is necessarily a polynomial ring. Further we show that separable ${\mathbb{A}}^2$-forms over PIDs are trivial.