Regular subcategories in bounded derived categories of affine schemes

Let $R$ be a commutative Noetherian ring such that $X=Spec R$ is connected. We prove that the category $D^b(coh X)$ contains no proper full triangulated subcategories which are regular. We also bound from below the dimension of a regular category $T$, if there exists a triangulated functor $T \to D^b(coh X)$ with certain properties. Applications are given to cohomological annihilator of $R$ and to point-like objects in $T$.

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