Stable connectivity over a base

Abstract

Morel's stable connectivity theorems state that for any connective $S^1$-spectrum $F$ of motivic spaces (Nisnevich simplicial sheaves) over an arbitrary field, the spectrum $L_{\mathbb A^1}(F)$ is connective, and the same property for $\mathbb P^1$-spectra of motivic spaces. Here $L_{\mathbb A^1}$ denotes the $\mathbb A^1$-localisation in the category of motivic spectra over a field $k$. Originally the same property was conjectured for the case of motivic $S^1$-spectra over a base scheme $S$.In view of Ayoub's conterexamples the modified version of conjecture states that $L_{\mathbb A^1}(F)$ is $(-d)$-connective for any connective $F$, where $d=\mathrm{dim} S$ is the Krull dimension. The conjecture is proven under the infiniteness assumption on the residue fields for the cases of Dedekind schemes by J.~Schmidt and F.~Strunk and noetherian domains of arbitrary dimension by N.~Deshmukh, A.~Hogadi, G.~Kulkarni and S.~Yadavand. In the article we prove the result or general base with out the assumption on the residue fields. So by the result for any smooth scheme $X$ over a base scheme $S$ of Krull dimension $d$ the Nisnevich sheaves of $S^1$-stable motivic homotopy groups $\pi_i^{S^1}(X)$ and $\mathbb P^1$-stable motivic homotopy groups $\pi_{i+j,j}^{\mathbb P^1}(X)$ vanishes for all $i<-d$.