Stable connectivity over a base

DOI10.1515/CRELLE-2022-0048arXiv1911.05014MaRDI QIDQ6329034

Anderi Eduardovich Druzhinin

Publication date: 12 November 2019

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 $d=mathrmdimS$ 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 $p{i}_i^{S^1}(X)$ and $mathbb{P}^1$-stable motivic homotopy groups $p{i}_{i+j,j}^{mathbb{P}^1}(X)$ vanishes for all $i<-d$.

Mathematics Subject Classification ID

Motivic cohomology; motivic homotopy theory (14F42)

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