A moving lemma for motivic spaces

Abstract:

The following moving lemma is proved. Let $k$ be a field and $X$ be a quasi-projective variety. Let $Z$ be a closed subset in $X$ and let $U$ be the semi-local scheme of finitely many closed points on $X$. Then the natural morphism $U\to X/(X-Z)$ of Nisnevich sheaves is $\mathbf {A}^1$-homotopic to the constant morphism of $U\to X/(X-Z)$ sending $U$ to the distinguished point of $X/(X-Z)$.