2020/03/15 by Aleksei Tsybyshev, Tsybyshev, Aleksei
Mathematics · #14F42 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2003.06892
openalex publication_date 2020/03/15 · openalex created_date 2020/03/23 · openalex updated_date 2026/07/28
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)→ Ω∞ℙ1 Σ∞ℙ1 (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))→ Ω∞ℙ1 Σ∞ℙ1 (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 ≥ 0, the simplicial sheaf C_*Fr((X,U)) is locally r-connected.