# Naive vs. genuine A^1-connectedness

## TL;DR

This paper investigates the differences between naive and genuine A^1-connectedness in algebraic geometry, revealing that trivial sections do not imply trivial sheaves and providing counterexamples related to A^1-locality.

## Contribution

It demonstrates that trivial sections of naive A^1-chain connected components do not guarantee triviality of genuine A^1-connected components, highlighting a key distinction.

## Key findings

- Trivial sections of naive A^1-chain components do not imply trivial genuine A^1-connectedness.
- Existence of an A^1-connected scheme where the Morel-Voevodsky singular construction is not A^1-local.
- Counterexamples to assumptions about A^1-connectedness properties.

## Abstract

We show that the triviality of sections of the sheaf of A^1-chain connected components of a space over finitely generated separable field extensions of the base field is not sufficient to ensure the triviality of the sheaf of its A^1-chain connected components, contrary to the situation with genuine A^1-connected components. As a consequence, we show that there exists an A^1-connected scheme for which the Morel-Voevodsky singular construction is not A^1-local.