TY - JOUR

T1 - Non-commutative localizations of additive categories and weight structures

AU - Bondarko, M.V.

AU - Sosnilo, V.A.

PY - 2018/9/1

Y1 - 2018/9/1

N2 - 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 by a set S of morphisms in the heart Hw of a weight structure w on it one obtains a triangulated category endowed with a weight structure w′. The heart of w′ is a certain version of the Karoubi envelope of the non-commutative localization Hw[S-1]add (of Hw by S). The functor Hw → Hw[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 S invertible. For any additive category A, taking C = Kb(A) we obtain a very ecient tool for computing A[S-1]add; using it, we generalize the calculations of Gerasimov and Malcolmson (made for rings only). We also prove that A[S-1]add coincides with the abstract localization A[S-1] (as constructed by Gabriel and Zisman) if S contains all identity morphisms of A and is closed with respect to direct sums. We apply our results to certain categories of birational motives DMo gm(U) (generalizing those defined by Kahn and Sujatha). We define DMo gm(U) for an arbitrary U as a certain localization of Kb(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.

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

KW - algebraic geometry

KW - birational motives

KW - category theory

KW - homological algebra

KW - K-theory

KW - non-commutative localizations

KW - t-structures

KW - triangulated categories

KW - weight structures

UR - http://www.scopus.com/inward/record.url?scp=84970003127&partnerID=8YFLogxK

U2 - 10.1017/S1474748016000207

DO - 10.1017/S1474748016000207

M3 - Article

VL - 17

SP - 785

EP - 821

JO - Journal of the Institute of Mathematics of Jussieu

JF - Journal of the Institute of Mathematics of Jussieu

SN - 1474-7480

IS - 4

ER -