Unstable étale motives

2025/07/27 by Mattis, Klaus
#Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #FOS: Mathematics

paper · doi:10.48550/arxiv.2507.20320

Abstract

We prove a rigidity result for certain p-complete étale A¹-invariant sheaves of anima over a qcqs finite-dimensional base scheme S of bounded étale cohomological dimension with p invertible on S. This generalizes results of Suslin--Voevodsky, Ayoub, Cisinski--Déglise, and Bachmann to the unstable setting. Over a perfect field we exhibit a large class of sheaves to which our main theorem applies, in particular the p-completion of the étale sheafification of any 2-effective 2-connective motivic space, as well as the p-completion of any 4-connective A¹-invariant étale sheaf. We use this rigidity result to prove (a weaker version of) an étale analog of Morel's theorem stating that for a Nisnevich sheaf of abelian groups, strong A¹-invariance implies strict A¹-invariance. Moreover, this allows us to construct an unstable étale realization functor on 2-effective 2-connective motivic spaces.

Related

• Hypercomplete étale framed motives and comparison of stable homotopy groups of motivic spectra and étale realizations over a field
• Unstable motivic and real-étale homotopy theory
• Strongly A¹-invariant sheaves (after F. Morel)
• Motivic sheaves revisited
• Surjectivity of the etale excision map for homotopy invariant framed presheaves