On the non-vanishing of local cohomology modules

It is shown that for any Artinian modules M, M∨ is the greatest integer i such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $${\text{H}}_m^i $$ \end{document}(M∨) ≠ 0.