Regular subcategories in bounded derived categories of affine schemes

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