(joint work with V. Sosnilo)

It is well known that for any ring commutative $R$ and a

multiplicative set of elements $S$ in it one can define the localized

ring $R[S^{-1}]$ of certain "fractions" (in particular, one obtains

the fraction field of an integral domain using this construction). If

$R$ is (associative unital but) not commutative then one cannot add

and multiply "fractions" of the form $r_1s_1^{-1}$ and $r_2s_2^{-1}$

to obtain a similar fraction in general. However, P.M. Conn has

proposed to look for initial ring homomorphisms $f:R\to R'$ that makes

all elements of $S$ invertible; one can also look for $f$ that make

certain homomorphisms between finitely generated projective (left)

$R$-modules invertible. This "non-commutative localization" setting

can be easily generalized to the search for a universal additive

functor that makes a set of morphisms $S$ in an additive category $A$

invertible. We have proved that this "additive localization" functor

is induced by the Verdier localization of the homotopy category

$K^b(A)$ by the subcategory generated by $Cone(S)$ (I will recall

these notions in my talk); this gave a natural generalization of

explicit descriptions of non-commutative localizations of rings (given

by Gerasimov, Malcolmson, and Schofield).

I hope to mention the relation of this argument to weight structures

and weight decompositions. This research was motivated by the study of

triangulated categories of birational motives (as defined by Kahn and

Sujatha), and I will gladly explain what these words mean to anybody

who is interested!

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