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NON-COMMUTATIVE LOCALIZATIONS OF ADDITIVE CATEGORIES AND WEIGHT STRUCTURES

Part of: Homological methods $K$-theory in geometry (Co)homology theory Cycles and subschemes Rings and algebras arising under various constructions Conditions on elements Homological algebra Abelian categories

Published online by Cambridge University Press:  24 May 2016

Mikhail V. Bondarko Open the ORCID record for Mikhail V. Bondarko [Opens in a new window]  and
Vladimir A. Sosnilo
Mikhail V. Bondarko
Affiliation:
St. Petersburg State University, Mathematics and Mechanics, St. Petersburg, Russian Federation (mbondarko@hotmail.com)
Vladimir A. Sosnilo
Affiliation:
St. Petersburg State University, Mathematics and Mechanics, St. Petersburg, Russian Federation (mbondarko@hotmail.com)
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Abstract

In this paper we demonstrate that non-commutative localizations of arbitrary additive categories (generalizing those defined by Cohn in the setting of rings) are closely (and naturally) related to weight structures. Localizing an arbitrary triangulated category $\text{}\underline{C}$ by a set $S$ of morphisms in the heart $\text{}\underline{Hw}$ of a weight structure $w$ on it one obtains a triangulated category endowed with a weight structure $w^{\prime }$. The heart of $w^{\prime }$ is a certain version of the Karoubi envelope of the non-commutative localization $\text{}\underline{Hw}[S^{-1}]_{\mathit{add}}$ (of $\text{}\underline{Hw}$ by $S$). The functor $\text{}\underline{Hw}\rightarrow \text{}\underline{Hw}[S^{-1}]_{\mathit{add}}$ is the natural categorical version of Cohn’s localization of a ring, i.e., it is universal among additive functors that make all elements of $S$ invertible. For any additive category $\text{}\underline{A}$, taking $\text{}\underline{C}=K^{b}(\text{}\underline{A})$ we obtain a very efficient tool for computing $\text{}\underline{A}[S^{-1}]_{\mathit{add}}$; using it, we generalize the calculations of Gerasimov and Malcolmson (made for rings only). We also prove that $\text{}\underline{A}[S^{-1}]_{\mathit{add}}$ coincides with the ‘abstract’ localization $\text{}\underline{A}[S^{-1}]$ (as constructed by Gabriel and Zisman) if $S$ contains all identity morphisms of $\text{}\underline{A}$ and is closed with respect to direct sums. We apply our results to certain categories of birational motives $DM_{gm}^{o}(U)$ (generalizing those defined by Kahn and Sujatha). We define $DM_{gm}^{o}(U)$ for an arbitrary $U$ as a certain localization of $K^{b}(Cor(U))$ and obtain a weight structure for it. When $U$ is the spectrum of a perfect field, the weight structure obtained this way is compatible with the corresponding Chow and Gersten weight structures defined by the first author in previous papers. For a general $U$ the result is completely new. The existence of the corresponding adjacent$t$-structure is also a new result over a general base scheme; its heart is a certain category of birational sheaves with transfers over $U$.

Keywords

category theoryhomological algebraK-theorynon-commutative localizationstriangulated categoriesweight structuresbirational motivest-structuresalgebraic geometry

MSC classification

Primary: 18E35: Localization of categories 16S85: Rings of fractions and localizations 14C15: (Equivariant) Chow groups and rings; motives
Secondary: 18E30: Derived categories, triangulated categories 18E05: Preadditive, additive categories 14F42: Motivic cohomology; motivic homotopy theory 19E15: Algebraic cycles and motivic cohomology 16U20: Ore rings, multiplicative sets, Ore localization 18G35: Chain complexes 16E20: Grothendieck groups, $K$-theory, etc.
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 17 , Issue 4 , September 2018 , pp. 785 - 821
Copyright
© Cambridge University Press 2016 

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Footnotes

Theorem 0.1 and Theorem 4.2.3 were proven under support of the Russian Science Foundation grant no. 14-21-00035. The research was also supported by RFBR grant no. 15-01-03034-a. Besides, the first author was supported by RFBR (grants no. 14-01-00393-a) and by Dmitry Zimin’s Foundation ‘Dynasty’; the second author was supported by the Chebyshev Laboratory (Department of Mathematics and Mechanics, St. Petersburg State University), by JSC ‘Gazprom Neft’, and by the Saint-Petersburg State University research grant no. 38.37.208.2016.

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