On Infinite Effectivity of Motivic Spectra and the Vanishing of their Motives
Mikhail Vladimirovich BondarkoSt. Petersburg State University, 14th Line 29B, Vasilyevsky Island, St. Petersburg 199178, Russia
We study the kernel of the "compact motivization" functor (i.e., we try to describe those compact objects of the -linear version of whose associated motives vanish; here . We also investigate the question when the -homotopy connectivity of ensures the -homotopy connectivity of itself (with respect to the homotopy -structure for . We prove that the kernel of vanishes and the corresponding "homotopy connectivity detection" statement is also valid if and only if is a non-orderable field; this is an easy consequence of similar results of T. Bachmann (who considered the case where the cohomological -dimension of is finite). Moreover, for an arbitrary the kernel in question does not contain any -torsion (and the author also suspects that all its elements are odd torsion unless . Furthermore, if the exponential characteristic of is invertible in then this kernel consists exactly of "infinitely effective" (in the sense of Voevodsky's slice filtration) objects of . 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.
Cite this article
Mikhail Vladimirovich Bondarko, On Infinite Effectivity of Motivic Spectra and the Vanishing of their Motives. Doc. Math. 25 (2020), pp. 811–840DOI 10.4171/DM/763