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 spacesis 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 * F r((X, U )) is Nisnevich-locally connected, and the motivic space morphism C * F r((X, U )) → Ω ∞ P 1 Σ ∞ P 1 (X/U ) is Nisnevichlocally a weak equivalence. Moreover, we show that if the codimension of S = X − U in each component of X is greater than r 0, the simplicial sheaf C * F r((X, U )) is locally r-connected.