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

This paper is dedicated to the study of the kernel of the “compact motivization” functor $M_k^c:S{H}^c(k)\to D{M}^c(k)$ (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 $M_k^c(E)$ 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 $S{H}^c(k)$ (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.