On Infinite Effectivity of Motivic Spectra and the Vanishing of their Motives

Abstract

We study the kernel of the "compact motivization" functor $M_{k,\Lambda}^c:SH^c_{\Lambda}(k)\to DM_{\Lambda}^c(k)$ (i.e., we try to describe those compact objects of the $\Lambda$-linear version of $SH(k)$ whose associated motives vanish; here $\mathbb{Z} \subset \Lambda \subset \mathbb{Q})$. We also investigate the question when the $0$-homotopy connectivity of $M^c_{k,\Lambda}(E)$ ensures the $0$-homotopy connectivity of $E$ itself (with respect to the homotopy $t$-structure $t_{\Lambda}^{SH}$ for $SH_{\Lambda}(k))$. We prove that the kernel of $M^c_{k,\Lambda}$ vanishes and the corresponding "homotopy connectivity detection" statement is also valid if and only if $k$ is a non-orderable field; this is an easy consequence of similar results of T. Bachmann (who considered the case where the cohomological $2$-dimension of $k$ is finite). Moreover, for an arbitrary $k$ the kernel in question does not contain any $2$-torsion (and the author also suspects that all its elements are odd torsion unless $\frac{1}{2}\in \Lambda)$. Furthermore, if the exponential characteristic of $k$ is invertible in $\Lambda$ then this kernel consists exactly of "infinitely effective" (in the sense of Voevodsky's slice filtration) objects of $SH^c_{\Lambda}(k)$. The results and methods of this paper are useful for the study of motivic spectra; they allow extending certain statements to motivic categories over direct limits of base fields. In particular, we deduce the tensor invertibility of motivic spectra of affine quadrics over arbitrary non-orderable fields from some other results of Bachmann. We also generalize a theorem of A. Asok.