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

TL;DR

This paper extends Faltings' Local-global Principle and Annihilator Theorems to derived categories, providing new variants for local cohomology modules of complexes with finitely generated homologies.

Contribution

It introduces derived category analogues of classical theorems, broadening their applicability to complexes with finitely generated homologies.

Findings

Established derived category versions of Faltings' Local-global Principle.

Proved derived Annihilator Theorems for local cohomology of complexes.

Provided variations of existing classical results in a derived setting.

Abstract

Let $\mathcal{Z}$ be a specialization closed subset of $\Spec R$ 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.