TY - JOUR

T1 - On Infinite Effectivity of Motivic Spectra and the Vanishing of their Motives

AU - Bondarko, Mikhail V.

PY - 2020/1/1

Y1 - 2020/1/1

N2 - We study the kernel of the “compact motivization” functor (formula presented) (i.e., we try to describe those compact objects of the A-linear version of SH(k) whose associated motives vanish; here ℤ ⊂ Λ ⊂ ℚ). We also investigate the question when the 0-homotopy connectivity of (formula presented) ensures the O-homotopy connectivity of E itself (with respect to the homotopy t-structure (formula presented) for SH^(k)). We prove that the kernel of (formula presented) vanishes and the corresponding “homotopy connectivity detection” statement is also valid if and only if A; 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 1/2 € A). Furthermore, if the exponential characteristic of k is invertible in A then this kernel consists exactly of “infinitely effective” (in the sense of Voe-vodsky’s slice filtration) objects of (formula presented). 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.

AB - We study the kernel of the “compact motivization” functor (formula presented) (i.e., we try to describe those compact objects of the A-linear version of SH(k) whose associated motives vanish; here ℤ ⊂ Λ ⊂ ℚ). We also investigate the question when the 0-homotopy connectivity of (formula presented) ensures the O-homotopy connectivity of E itself (with respect to the homotopy t-structure (formula presented) for SH^(k)). We prove that the kernel of (formula presented) vanishes and the corresponding “homotopy connectivity detection” statement is also valid if and only if A; 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 1/2 € A). Furthermore, if the exponential characteristic of k is invertible in A then this kernel consists exactly of “infinitely effective” (in the sense of Voe-vodsky’s slice filtration) objects of (formula presented). 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.

KW - connectivity

KW - conservativity

KW - continuity

KW - homotopy t-structures

KW - infinite effectivity

KW - motives

KW - Motivic stable homotopy category

KW - motivization

UR - http://www.scopus.com/inward/record.url?scp=85104676889&partnerID=8YFLogxK

U2 - 10.25537/dm.2020v25.811-840

DO - 10.25537/dm.2020v25.811-840

M3 - Article

AN - SCOPUS:85104676889

VL - 25

SP - 811

EP - 840

JO - Documenta Mathematica

JF - Documenta Mathematica

SN - 1431-0635

ER -