Associated primes of local cohomology module

Abstract

Let $\fa$ be an ideal of a commutative Noetherian ring $R$ and $M$ a finitely generated $R$-module. Let $t$ be a natural integer. It is shown that there is a finite subset $X$ of $\Spec R$, such that $\Ass_R(H_{\fa}^t(M))$ is contained in $X$ union with the union of the sets $\Ass_R(\Ext_R^j(R/\fa,H_{\fa}^i(M)))$, where $0\leq i<t$ and $0\leq j\leq t^2+1$. As an immediate consequence, we deduce that the first non $\fa$-cofinite local cohomology module of $M$ with respect to $\fa$ has only finitely many associated prime ideals.