We study the kernel of the ``compact motivization'' functor (M_{k,Lambda}^c:SH^c_{Lambda}(k)o 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 (rac{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.
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