A motivic Segal theorem for open pairs of smooth schemes over an infinite perfect field

Abstract: V. Voevodsyky laid the groundwork of delooping motivic spaces in order to provide a new, more computation-friendly, construction of the stable motivic category $SH(k)$, G. Garkusha and I. Panin made that project a reality, while collaborating with A. Ananievsky, A. Neshitov and A. Druzhinin. In particular, G. Garkusha and I. Panin proved that for an infinite perfect field $k$ and any $k$-smooth scheme $X$ the canonical morphism of motivic spaces $C_*Fr(X)\to \Omega^{\infty}_{\mathbb{P}^1} \Sigma^{\infty}_{\mathbb{P}^1} (X_+)$ is Nisnevich-locally a group-completion. In the present work, a generalisation of that theorem to the case of smooth open pairs $(X,U),$ where $X$ is a $k$-smooth scheme, $U$ is its open subscheme intersecting each component of $X$ in a nonempty subscheme. We claim that in this case the motivic space $C_*Fr((X,U))$ is Nisnevich-locally connected, and the motivic space morphism $C_*Fr((X,U))\to \Omega^{\infty}_{\mathbb{P}^1} \Sigma^{\infty}_{\mathbb{P}^1} (X/U)$ is Nisnevich-locally a weak equivalence. Moreover, we show that if the codimension of $S=X-U$ in each component of $X$ is greater than $r \geq 0,$ the simplicial sheaf $C_*Fr((X,U))$ is locally $r$-connected.