NON-COMMUTATIVE LOCALIZATIONS OF ADDITIVE CATEGORIES AND WEIGHT STRUCTURES

作者：Bondarko Mikhail V^*; Sosnilo Vladimir A

来源：Journal of the Institute of Mathematics of Jussieu, 2018, 17(4): 785-821.

DOI：10.1017/S1474748016000207

• category theory
• homological algebra
• K-theory
• non-commutative localizations
• triangulated categories
• weight structures
• birational motives
• t-structures
• algebraic geometry

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 (C) under bar by a set of morphisms in the heart (Hw) under bar of a weight structure on it one obtains a triangulated category endowed with a weight structure. The heart of w' is a certain version of the Karoubi envelope of the non-commutative localization (Hw) under bar [S-1](add) (of (Hw) under bar by). The functor (Hw) under bar -> (Hw) under bar [S-1](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 invertible. For any additive category, (A) under bar taking (C) under bar = K-b((A) under bar) we obtain a very efficient tool for computing; (A) under bar [S-1](add) using it, we generalize the calculations of Gerasimov and Malcolmson (made for rings only). We also prove that (A) under bar [S-1](add) coincides with the 'abstract' localization (A) under bar [S-1] (as constructed by Gabriel and Zisman) if contains all identity morphisms (A) under bar of and is closed with respect to direct sums. We apply our results to certain categories of birational motives DMgmo (U) (generalizing those defined by Kahn and Sujatha). We define DMgmo(U) for an arbitrary 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 -structure is also a new result over a general base scheme; its heart is a certain category of birational sheaves with transfers over U.

• 出版日期2018-9