Some results on local cohomology modules defined by a pair of ideals

Abstract

Let $R$ be a commutative Noetherian ring, and let $I$ and $J$ be two ideals of $R$. Assume that $R$ is local with the maximal ideal ${\mathfrak{m}}$, we mainly prove that (i) there exists an equality \[{\text{inf}}\{i\, \mid H_{I,J}^i(M)\, {\text{ is not Artinian}} \}={\text{inf}}\{ {\text{depth}}M_{\mathfrak{p}} \mid \, {\mathfrak{p}}\in W(I, J)\backslash \{{\mathfrak{m}}\} \}\] for any finitely generated $R-$module $M$, where $W(I, J)=\{{\mathfrak{p}} \in {\text{Spec}}(R) \mid \, I^n \subseteq {\mathfrak{p}}+J\,\, {\text{for some positive integer}} \,n \}$; (ii) for any finitely generated $R-$module $M$ with ${\text{dim}}M=d$, $H_{I,J}^d(M)$ is Artinian. Also, we give a characterization to the supremum of all integers $r$ for which $H_{I,J}^r(M) \neq 0$.