The derived category analogue of the Hartshorne-Lichtenbaum Vanishing Theorem

TL;DR

This paper extends the Hartshorne-Lichtenbaum Vanishing Theorem to the derived category setting, establishing conditions for the vanishing of local cohomology of complexes in terms of prime ideal dimensions.

Contribution

It introduces a derived category analogue of the classical vanishing theorem, connecting local cohomology vanishing to prime ideal dimension conditions in the completed ring.

Findings

Characterizes vanishing of local cohomology for complexes.

Links vanishing to prime ideal dimension inequalities.

Provides a criterion involving the completion of the ring.

Abstract

Let $\fa$ be an ideal of a local ring $(R,\fm)$ and $X$ a $d$-dimensional homologically bounded complex of $R$-modules whose all homology modules are finitely generated. We show that $H^d_{\fa}(X)=0$ if and only if $\dim \hat{R}/\fa \hat{R}+\fp>0$ for all prime ideals $\fp$ of $\hat{R}$ such that $\dim \hat{R}/\fp-\inf (X\otimes_R\hat{R})_{\fp}=d$.