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Let $\mathcal{Z}$ be a specialization closed subset of $\Spec R$ and $X$ a left homologically bounded complex
with finitely generated homologies. We provide some inequalities between the Bass numbers of $X$ and its local cohomology modules with respect to $\mathcal{Z}$. As an application of these inequalities we provide a comparison between the injective dimensions of $X$ and its non-zero local cohomology module $\H_{\mathcal{Z}}^i(X)$. Our versions contain variations of some results already known on these area.
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