On localizing rings, categories, and weight structures

(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!