The Derived Category Analogue of the Hartshorne-Lichtenbaum Vanishing Theorem

Abstract

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