${\mathbb{A}}^1$-connectivity of motivic spaces

Abstract

We prove a version of Morel's unstable ${\mathbb{A}}^1$-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.

arXiv commentVarious minor changes. To appear in J. Inst. Math. Jussieu