On Infinite Effectivity of Motivic Spectra and the Vanishing of their Motives

Mikhail Vladimirovich Bondarko

Documenta mathematica, Tome 25 (2020), pp. 811-840

Résumé

We study the kernel of the "compact motivization" functor Mk,Λc​:SHΛc​(k)→DMΛc​(k) (i.e., we try to describe those compact objects of the Λ-linear version of SH(k) whose associated motives vanish; here Z⊂Λ⊂Q). We also investigate the question when the 0-homotopy connectivity of Mk,Λc​(E) ensures the 0-homotopy connectivity of E itself (with respect to the homotopy t-structure tΛSH​ for SHΛ​(k)). We prove that the kernel of Mk,Λc​ 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 21​∈Λ). Furthermore, if the exponential characteristic of k is invertible in Λ then this kernel consists exactly of "infinitely effective" (in the sense of Voevodsky's slice filtration) objects of SHΛc​(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.

Détail

DOI : 10.4171/dm/763

Classification : 11E81, 14C15, 14F20, 14F42, 18E40, 18G80, 55P42
Mots-clés : motives, motivic stable homotopy category, motivization, conservativity, infinite effectivity, connectivity, homotopy t-structures, continuity