\mathbbA¹-connectivity of motivic spaces

2025/12/11 by Tess Bouis, Bouis, Tess, Arnab Kundu +1
Computer Science · Mathematics · #Homotopy and Cohomology in Algebraic Topology #Polynomial and algebraic computation #Topological and Geometric Data Analysis #math.AG #math.AT #math.KT

paper · pdf · doi:10.48550/arxiv.2512.10712

Various minor changes. To appear in J. Inst. Math. Jussieu

openalex publication_date 2025/12/11 · openalex created_date 2025/12/13 · arxiv created 2026/07/28 · arxiv updated 2026/07/30 · openalex updated_date 2026/08/01

Abstract

We prove a version of Morel's unstable \mathbbA¹-connectivity theorem over arbitrary base schemes. In the stable setting, this recovers (and simplifies the proof of) the known connectivity bounds due to Morel, Schmidt--Strunk, Deshmukh--Hogadi--Kulkarni--Yadav, and Druzhinin, and extends them to possibly non-noetherian schemes. Using the recent work of Bachmann--Elmanto--Morrow, this also implies that the slice filtration on homotopy K-theory is convergent for qcqs schemes of finite valuative dimension.

Citations

• \mathbbA¹-invariant motivic cohomology of schemes
• Beilinson--Lichtenbaum phenomenon for motivic cohomology
• Gabber presentation lemma over noetherian domains
• Dimensional homotopy t-structure in motivic homotopy theory

Related

• Stable connectivity over a base
• On P¹-stabilization in unstable motivic homotopy theory
• The trivial fiber topology and framed motives over the integers
• Stable motivic trivial fibre decomposition over integers and framed motives
• The motivic Hopf map solves the homotopy limit problem for K-theory