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Suppose I have a smooth projective variety $X$, and a semi-orthogonal decomposition of its bounded derived category:

$$D^b(X)= < A, E_1, E_2, ... , E_n >$$

where the $E_i$ are fully faithful, saturated (and hence admissible) subcats of $D^b(X)$. Is A authomatically saturated/admissible? Why? If not, under what assuptions?

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Yes, it is. See Bondal, A. I.; Kapranov, M. M. Representable functors, Serre functors, and reconstructions.

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