Volume 37 • Number 1 • 2014   • A Non-Vanishing Theorem for Local Cohomology Modules Amir Bagheri Abstract. Assume that $(R,m)$ a local Noetherian ring and $a$ is an ideal of $R$. In this paper we introduce a new class of $R$-modules denoted by weakly finite modules that is a generalization of finitely generated modules and containing the class of Big Cohen-Macaulay modules and $a$-cofinite modules. We improve the non-vanishing theorem due to Grothendieck for weakly finite modules. Finally we define the notion depth$_{R}M$ and we prove that if $M$ is a weakly finite $R$-module and $H_{m}^{i}(M)\neq 0$ for some $i$, then depth$_{R}M$$\leq i \leq \dim M$. 2010 Mathematics Subject Classification: 13D45 Full text: PDF