A motivic Segal-type theorem for pairs (announcement)

A. Tsybyshev

Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 35, Tome 484 (2019), pp. 165-184

Résumé

V. Voevodsky has set the foundation of the machinery of loop spaces of motivic spaces to provide a more computation-friendly construction of the stable motivic category $SH(k)$. G. Garkusha and I. Panin have made that vision a reality, using joint works with A. Ananievsky, A. Neshitov and A. Druzhinin. In particular, G. Garkusha and I. Panin have proved that for any 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. The present work addresses a generalisation of that theorem to the case of general open pairs of smooth schemes $(X,U),$ where $X$ is a $k$-smooth scheme, $U$ is its open subscheme intersecting each component of $X$ in a nonempty subscheme. We propose that in this case the motivic space $C_*Fr((X,U))$ is Nisnevich-locally connected and the canonical morphism of motivic spaces $C_*Fr((X,U))\to \Omega^{\infty}_{\mathbb{P}^1} \Sigma^{\infty}_{\mathbb{P}^1} (X/U)$ is Nisnevich-locally a homotopy equivalence of simplicial sets. Moreover, we state that if the codimension of $S=X-U$ in each component of $X$ is greater than $r \geq 0,$ then the simplicial sheaf $C_*Fr((X,U))$ is locally $r$-connected. Some principal steps of the proof of these statements are provided in the present paper, but other important technical lemmas are given without proof. Those proofs will be published later.