Naive vs. genuine A1-connectedness
Abstract
We show that the triviality of sections of the sheaf of A1-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 A1-chain connected components, contrary to the situation with genuine A1-connected components. As a consequence, we show that there exists an A1-connected scheme for which the Morel-Voevodsky singular construction is not A1-local.
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