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Abstract

Let ${\mit\Lambda} $ be an artin algebra. We prove that for each sequence $(h_{i})_{i\in \mathbb{Z}}$ of non-negative integers there are only a finite number of isomorphism classes of indecomposables $X\in \mathcal{D}^{\rm b}({\mit\Lambda} ),$ the bounded derived category of ${\mit\Lambda} $, with $\mathop{\rm length}\nolimits _{E(X)}H^{i}(X)=h_{i}$ for all $i\in \mathbb{Z}$ and $E(X)$ the endomorphism ring of $X$ in $\mathcal{D}^{\rm b}({\mit\Lambda} )$ if and only if $\mathcal{D}^{\rm b}(\mathop{\rm Mod}\nolimits {\mit\Lambda} )$, the bounded derived category of the category $\mathop{\rm Mod}\nolimits {\mit\Lambda} $ of all left ${\mit\Lambda} $-modules, has no generic objects in the sense of [4].