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 Mk,Λc:SHΛc(k)DMΛc(k)M_{k,\Lambda}^c:SH^c_{\Lambda}(k)\to DM_{\Lambda}^c(k) (i.e., we try to describe those compact objects of the Λ\Lambda-linear version of SH(k)SH(k) whose associated motives vanish; here ZΛQ)\mathbb{Z} \subset \Lambda \subset \mathbb{Q}). We also investigate the question when the 00-homotopy connectivity of Mk,Λc(E)M^c_{k,\Lambda}(E) ensures the 00-homotopy connectivity of EE itself (with respect to the homotopy tt-structure tΛSHt_{\Lambda}^{SH} for SHΛ(k))SH_{\Lambda}(k)). We prove that the kernel of Mk,ΛcM^c_{k,\Lambda} vanishes and the corresponding "homotopy connectivity detection" statement is also valid if and only if kk is a non-orderable field; this is an easy consequence of similar results of T. Bachmann (who considered the case where the cohomological 22-dimension of kk is finite). Moreover, for an arbitrary kk the kernel in question does not contain any 22-torsion (and the author also suspects that all its elements are odd torsion unless 12Λ)\frac{1}{2}\in \Lambda). Furthermore, if the exponential characteristic of kk is invertible in Λ\Lambda then this kernel consists exactly of "infinitely effective" (in the sense of Voevodsky's slice filtration) objects of SHΛc(k)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.

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