Non-commutative localizations of additive categories and weight structures; applications to birational motives

Abstract

In this paper we demonstrate that 'non-commutative localizations' of arbitrary additive categories (generalizing those defined by Cohn for rings) are closely (and naturally) related with weight structures. Localizing an arbitrary triangulated $C$ by a set $S$ of morphisms in the heart of a weight structure $w$ for it one obtains a triangulated category endowed with a weight structure $w'$. The heart of $w'$ is a certain idempotent completion of the non-commutative localization of the heart of $w$ by $S$. The latter is the natural categorical version of Cohn's localizations of rings i.e. the functor connecting hearts is universal among all the additive functors that make the elements of $S$ invertible. In particular, taking $C=K^b(A)$ for an additive $A$ we obtain a very efficient tool for computing the additive localization of $A$ by $S$; using it, we generalize the calculations of Gerasimov and Malcolmson. We apply our results to certain categories of birational motives over a base scheme $U$ (generalizing those defined by Kahn and Sujatha). When $U$ is the spectrum of a perfect field, the weight structure obtained is compatible with the Chow and Gersten weight structures defined by the first author in previous papers. For a general $U$ the result is completely new. We also consider the relation of weight structures with their adjacent t-structures (in localizations). In the 'motivic' setting mentioned this yields the natural generalization of the 'duality' between birational motives and birational sheaves with transfers established by Kahn and Sujatha.