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

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.

8 pages, comments are welcome; v2: minor modifications

References in corpus (1)

• A^1-connected components of ruled surfaces