Remarks on iterations of the $\mathbb A^1$-chain connected components construction

TL;DR

This paper investigates the behavior of the sheaf of a1^1-connected components, showing it aligns with its universal quotient on field points and providing explicit formulas, while also constructing examples where iterative processes do not stabilize early.

Contribution

It establishes the equivalence of the sheaf of a1^1-connected components with its universal quotient on field points and constructs examples demonstrating non-stabilization of iterations.

Findings

Sheaf of a1^1-connected components matches its universal quotient on field points.

Explicit formula for field-valued points of a1^1-connected components.

Existence of spaces where iterative a1^1-connected components do not stabilize before stage n.

Abstract

We show that the sheaf of $\mathbb A^1$-connected components of a Nisnevich sheaf of sets and its universal $\mathbb A^1$-invariant quotient (obtained by iterating the $\mathbb A^1$-chain connected components construction and taking the direct limit) agree on field-valued points. This establishes an explicit formula for the field-valued points of the sheaf of $\mathbb A^1$-connected components of any space. Given any natural number $n$, we construct an $\mathbb A^1$-connected space on which the iterations of the naive $\mathbb A^1$-connected components construction do not stabilize before the $n$th stage.