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

Let $\mathcal{Z}$ be a specialization closed subset of $\text{Spec}R$ and $X$ a homologically left-bounded complex with finitely generated homologies. We establish Faltings’ Local-global Principle and Annihilator Theorems for the local cohomology modules {{\H_{\mathcal{Z}}^i(X). }} Our versions contain variations of results already known on these theorems.