On Infinite Effectivity of Motivic Spectra and the Vanishing of their Motives
Mikhail Vladimirovich Bondarko
St. Petersburg State University, 14th Line 29B, Vasilyevsky Island, St. Petersburg 199178, Russia
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Abstract
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. 811–840
DOI 10.4171/DM/763