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Let $B$ be an additive category (a small one; one can assume that it is a $\mathbb{Q}$-category, yet not much else is known about it). Given a set of objects $S$ of $K^b(B)$ (or $K(B)$), I consider the localization (i.e. the Verdier quotient) of $K^b(B)$ by the full triangulated subcategory generated by $S$. My question is: are there any clever ways for computing morphism groups in this localization. I know of two ones:

  1. The calculus of fractions.

  2. Drinfeld's construction of localizations of triangulated categories endowed with differential graded enhancements (here a certain flatness restriction on morphisms is needed; $\mathbb{Q}$-coefficients are sufficient for that).

Are there any other ways known for 'nice' $S$? As a beginning, it would be nice to have a comprehencive understanding of the case $S\subset B$.

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