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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
by a set
of morphisms in the heart
of a weight structure
on it one obtains a triangulated category endowed with a weight structure
. The heart of
is a certain version of the Karoubi envelope of the non-commutative localization
). The functor
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
we obtain a very efficient tool for computing
; using it, we generalize the calculations of Gerasimov and Malcolmson (made for rings only). We also prove that
coincides with the ‘abstract’ localization
(as constructed by Gabriel and Zisman) if
contains all identity morphisms of
and is closed with respect to direct sums. We apply our results to certain categories of birational motives
(generalizing those defined by Kahn and Sujatha). We define
for an arbitrary
as a certain localization of
and obtain a weight structure for it. When
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
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
The goal of this paper is to prove that if certain ‘standard’ conjectures on motives over algebraically closed fields hold, then over any ‘reasonable’ scheme
there exists a motivic
-structure for the category
of relative Voevodsky’s motives (to be more precise, for the Beilinson motives described by Cisinski and Deglise). If
is of finite type over a field, then the heart of this
-structure (the category of mixed motivic sheaves over
) is endowed with a weight filtration with semisimple factors. We also prove a certain ‘motivic decomposition theorem’ (assuming the conjectures mentioned) and characterize semisimple motivic sheaves over
in terms of those over its residue fields. Our main tool is the theory of weight structures. We actually prove somewhat more than the existence of a weight filtration for mixed motivic sheaves: we prove that the motivic
-structure is transversal to the Chow weight structure for
(that was introduced previously by Hébert and the author). We also deduce several properties of mixed motivic sheaves from this fact. Our reasoning relies on the degeneration of Chow weight spectral sequences for ‘perverse étale homology’ (which we prove unconditionally); this statement also yields the existence of the Chow weight filtration for such (co)homology that is strictly restricted by (‘motivic’) morphisms.
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