On infinite effectivity of motivic spectra and the vanishing of their motives

TL;DR

This paper investigates the kernel of the motivization functor in motivic homotopy theory, proving it vanishes over non-orderable fields and characterizing its elements as infinitely effective objects, with implications for invertibility of motives.

Contribution

It establishes the vanishing of the kernel of the compact motivization functor over non-orderable fields and characterizes its elements as infinitely effective, extending results on motivic spectra.

Findings

Kernel of motivization functor vanishes over non-orderable fields

Kernel contains no 2-torsion, likely only odd torsion

Elements are exactly infinitely effective objects in the slice filtration

Abstract

This paper is dedicated to the study of the kernel of the "compact motivization" functor $M_{k}^c:SH^c(k)\to DM^c(k)$ (i.e., we try to describe those compact objects of $SH(k)$ whose associated motives vanish. Moreover, we study the question when the $m$-connectivity of $M^c_{k}(E)$ ensures the $m$-connectivity of $E$ itself (with respect to the corresponding homotopy t-structures). We prove that the kernel of $M_{k}^c$ vanishes and the corresponding "connectivity detection" statement is also valid if and only if $k$ is a non-orderable field; this is an easy consequence of the corresponding results of T. Bachmann (who considered the case where the $2$-adic cohomological dimension of $k$ is finite). We also sketch a deduction of these statements from the "slice-convergence" results of M. Levine. Moreover, for a general $k$ we prove that this kernel does not contain any $2$-torsion; the author also suspects that all its elements are odd torsion. Besides we prove that the kernel in question consists exactly of "infinitely effective" (in the sense of Voevodsky's slice filtration) objects of $SH^c(k)$ (assuming that the exponential characteristic of $k$ is inverted in the coefficient ring). These result allow (following another idea of Bachmann) to carry over his results on the tensor invertibility of certain motives of affine quadrics to the corresponding motivic spectra whenever $k$ is non-orderable. We also generalize a theorem of A. Asok.