TY - JOUR

T1 - On the Weight Lifting Property for Localizations of Triangulated Categories

AU - Bondarko, M. V.

AU - Sosnilo, V. A.

PY - 2018/9/1

Y1 - 2018/9/1

N2 - As we proved earlier, for any triangulated category C_ endowed with a weight structure w and a triangulated subcategory D_ of C_ (strongly) generated by cones of a set of morphism S in the heart Hw_ of w there exists a weight structure w' on the Verdier quotient C'_=C_/D_ such that the localization functor C_→C'_ is weight-exact (i.e., “respects weights”). The goal of this paper is to find conditions ensuring that for any object of C'_ of non-negative (resp. non-positive) weights there exists its preimage in C_ satisfying the same condition; we call a certain stronger version of the latter assumption the left (resp., right) weight lifting property. We prove that that these weight lifting properties are fulfilled whenever the set S satisfies the corresponding (left or right) Ore conditions. Moreover, if D_ is generated by objects of Hw_ then any object of Hw'_ lifts to Hw_. We apply these results to obtain some new results on Tate motives and finite spectra (in the stable homotopy category). Our results are also applied to the study of the so-called Chow-weight homology in another paper.

AB - As we proved earlier, for any triangulated category C_ endowed with a weight structure w and a triangulated subcategory D_ of C_ (strongly) generated by cones of a set of morphism S in the heart Hw_ of w there exists a weight structure w' on the Verdier quotient C'_=C_/D_ such that the localization functor C_→C'_ is weight-exact (i.e., “respects weights”). The goal of this paper is to find conditions ensuring that for any object of C'_ of non-negative (resp. non-positive) weights there exists its preimage in C_ satisfying the same condition; we call a certain stronger version of the latter assumption the left (resp., right) weight lifting property. We prove that that these weight lifting properties are fulfilled whenever the set S satisfies the corresponding (left or right) Ore conditions. Moreover, if D_ is generated by objects of Hw_ then any object of Hw'_ lifts to Hw_. We apply these results to obtain some new results on Tate motives and finite spectra (in the stable homotopy category). Our results are also applied to the study of the so-called Chow-weight homology in another paper.

KW - idempotent completions

KW - localizations

KW - spectra

KW - stable homotopy category

KW - triangulated categories

KW - Voevodsky motives

KW - Weight structures

KW - T-STRUCTURES

KW - MOTIVES

UR - http://www.scopus.com/inward/record.url?scp=85053561385&partnerID=8YFLogxK

UR - http://arxiv.org/abs/1510.03403

UR - http://www.mendeley.com/research/weight-lifting-property-localizations-triangulated-categories

U2 - 10.1134/S1995080218070077

DO - 10.1134/S1995080218070077

M3 - Article

AN - SCOPUS:85053561385

VL - 39

SP - 970

EP - 984

JO - Lobachevskii Journal of Mathematics

JF - Lobachevskii Journal of Mathematics

SN - 1995-0802

IS - 7

ER -