arXiv subjects

Mikhail Bondarko

Publications and source records attributed to Mikhail Bondarko.

3 recordsLinked to original sources

arXiv2017

On the weight lifting property for localizations of triangulated categories

As we proved earlier, for a triangulated category $\underline{C}$ endowed with a weight structure $w$ and a triangulated subcategory $\underline{D}$ of $\underline{C}$ (strongly) generated by cones of a set of morphisms $S$ in the heart $\underline{Hw}$ of $w$ there exists a weight structure $w'$ on the Verdier quotient $\underline{C}'=\underline{C}/\underline{D}$ such that the localization functor $\underline{C} \to \underline{C}'$ is weight-exact (i.e., "respects weights"). The goal of this paper is to find conditions ensuring that for any object of $\underline{C}'$ of non-negative (resp. non-positive) weights there exists its preimage in $\underline{C}$ satisfying the same condition; we call a certain stronger version of the latter assumption the left (resp., right) weight lifting property. We prove that these weight lifting properties are fulfilled whenever the set $S$ satisfies the corresponding (left or right) Ore conditions. Moreover, if $\underline{D}$ is generated by objects of $\underline{Hw}$ then any object of $\underline{Hw}'$ lifts to $\underline{Hw}$. 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.

math.KT↗

arXiv2016

Dimensional homotopy t-structure in motivic homotopy theory

The aim of this work is to construct certain homotopy t-structures on various categories of motivic homotopy theory, extending works of Voevodsky, Morel, Déglise and Ayoub. We prove these $t$-structures possess many good properties, some analogous to that of the perverse $t$-structure of Beilinson, Bernstein and Deligne. We compute the homology of certain motives, notably in the case of relative curves. We also show that their hearts provide convenient extensions of the theory of homotopy invariant sheaves with transfers, extending some of the main results of Voevodsky. These t-structures are closely related to Gersten weight structures as defined by Bondarko.

math.AG↗

arXiv2016

Picard groups, weight structures, and (noncommutative) mixed motives

We develop a general theory which enables the computation of the Picard group of a symmetric monoidal triangulated category, equipped with a weight structure, in terms of the Picard group of the associated heart. As an application, we compute the Picard group of several categories of motivic nature - mixed Artin motives, mixed Artin-Tate motives, motivic spectra, noncommutative mixed Artin motives, noncommutative mixed motives of central simple algebras, noncommutative mixed motives of separable algebras - as well as the Picard group of the derived categories of symmetric ring spectra.

math.AG↗