Annihilation of cohomology and decompositions of derived categories

It is proved that an element $r$ in the center of a coherent ring $\mathrm{\Lambda }$ annihilates ${\mathrm{E}\mathrm{x}\mathrm{t}}_{\mathrm{\Lambda }}^n(M,N)$, for some positive integer $n$ and all finitely presented $\mathrm{\Lambda }$-modules $M$ and $N$, if and only if the bounded derived category of $\mathrm{\Lambda }$ is an extension of the subcategory consisting of complexes annihilated by $r$ and those obtained as $n$-fold extensions of $\mathrm{\Lambda }$. This has applications to finiteness of dimension of derived categories.