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Let $\mathcal{Z}$ be a specialization closed subset of $\Spec R$ and $X$ ahomologically left-bounded complex with finitely generated homologies. Weestablish Faltings' Local-global Principle and Annihilator Theorems for thelocal cohomology modules {{$\H_{\mathcal{Z}}^i(X).$ }} Our versions containvariations of results already known on these theorems.

The derived category analogues of Faltings' Local-global Principle and Annihilator Theorems