Associated primes of local cohomology modules

Abstract:

Let $\mathfrak {a}$ 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 $\operatorname {Spec}R$, such that $\operatorname {Ass}_R(H_{\mathfrak {a}}^t(M))$ is contained in $X$ union with the union of the sets $\operatorname {Ass}_R(\operatorname {Ext} _R^j(R/\mathfrak {a},H_{\mathfrak {a}}^i(M)))$, where $0\leq i<t$ and $0\leq j\leq t^2+1$. As an immediate consequence, we deduce that the first non-$\mathfrak {a}$-cofinite local cohomology module of $M$ with respect to $\mathfrak {a}$ has only finitely many associated prime ideals.