On codimension-one $\mathbb A^1$-fibration with retraction

TL;DR

This paper investigates conditions under which flat algebras with a retraction are locally $ ext{A}^1$-fibrations, focusing on codimension-one fiber properties to establish sufficiency criteria.

Contribution

It provides new results on when codimension-one fiber conditions guarantee that a flat algebra with a retraction is locally an $ ext{A}^1$-fibration.

Findings

Identifies sufficient fiber conditions for $ ext{A}^1$-fibration structures.

Establishes criteria for flat algebras with retractions to be locally $ ext{A}^1$-fibrations.

Advances understanding of algebraic structures with $ ext{A}^1$-fibrations.

Abstract

In this paper we prove some results on the sufficiency of codimension-one fibre conditions for a flat algebra with a retraction to be locally $\mathbb{A}^1$ or at least an $\mathbb{A}^1$-fibration.