A Motivic Segal Theorem for Pairs (Announcement)

Abstract

In order to provide a new, more computation-friendly, construction of the stable motivic category SH(k), V. Voevodsyky laid the foundation of delooping motivic spaces. G. Garkusha and I. Panin based on joint works with A. Ananievsky, A. Neshitov, and A. Druzhinin made that project a reality. 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 $$ {C}_{\ast } Fr(X)\to {\Omega}_{{\mathrm{\mathbb{P}}}^1}^{\infty }{\sum}_{{\mathrm{\mathbb{P}}}^1}^{\infty}\left({X}_{+}\right) $$ is a Nisnevich locally group-completion. In the present paper, a generalization of that theorem is established to the case of smooth open pairs (X,U), where X is a k-smooth scheme and U is its open subscheme intersecting each component of X in a nonempty subscheme. It is claimed that in this case the motivic space C*Fr((X,U)) is a Nisnevich locally connected, and the motivic space morphism $$ {C}_{\ast } Fr\left(\left(X,U\right)\right)\to {\Omega}_{{\mathrm{\mathbb{P}}}^1}^{\infty }{\sum}_{{\mathrm{\mathbb{P}}}^1}^{\infty}\left(X/U\right) $$ is Nisnevich locally weak equivalence. Moreover, it is proved that if the codimension of S = X−U in each component of X is greater than r ≥ 0, then the simplicial sheaf C*Fr((X,U)) is locally r-connected.