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We introduce the notion of length of triangulated categories, and we compute the length of the derived category D^ b (C) of a smooth projective curve C and classify all finite length thick subcategories of D^ b (C). We also discuss the length of (1) derived categories of finite dimensional representations of ADE quivers, (2) derived categories of some singular varieties and (3) Krah's phantom subcategories.
Hirano et al. (Thu,) studied this question.
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