Remarks on iterations of the A1-chain connected components construction.

Abstract

We show that the sheaf of A 1 - connected components of a Nisnevich sheaf of sets and its universal A 1 - invariant quotient (obtained by iterating the 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 A 1 - connected components of any space. Given any natural number n , we construct an A 1 -connected space on which the iterations of the naive A1 - connected components construction do not stabilize before the n-th stage.