On Infinite Effectivity of Motivic Spectra and the Vanishing of their Motives
Mikhail Vladimirovich Bondarko · Documenta Mathematica · 2020
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.