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

TL;DR

This paper establishes an unstable version of Morel's $A^1$-connectivity theorem over arbitrary base schemes, extending known results to non-noetherian schemes and implications for homotopy $K$-theory.

Contribution

It proves an unstable $A^1$-connectivity theorem over arbitrary schemes, generalizing and simplifying existing stable results and linking to slice filtration convergence in homotopy $K$-theory.

Findings

Proves unstable $A^1$-connectivity over arbitrary schemes.

Extends stable connectivity bounds to non-noetherian schemes.

Shows convergence of slice filtration on homotopy $K$-theory for schemes of finite valuative dimension.

Abstract

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