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

TL;DR

This paper proves that certain algebraic fibrations over discrete valuation rings and PIDs are necessarily trivial, specifically showing they are polynomial rings or trivial forms.

Contribution

It establishes the triviality of $A^2$-fibrations over DVRs and PIDs, extending understanding of algebraic structures over these rings.

Findings

Any $A^2$-fibration over a DVR that is an $A^2$-form is a polynomial ring.

Separable $A^2$-forms over PIDs are trivial.

The results generalize the triviality of algebraic fibrations over specific rings.

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.