Bass numbers of local cohomology modules with respect a pair of ideals

Abstract: Let $R$ be a Noetherian local ring, $I$ and $J$ two ideals of $R$, $M$ an $R$-module and $s$ and $t$ two integers. We study the relationship between the Bass numbers of $M$ and $H^{i}_{I,J}(M)$. We show that $\mu^t(M)\leq\sum_{i=0}^{t}\mu^{t-i}(H^{i}_{I,J}(M))$ and $\mu^s(H^{t}_{I,J}(M))\leq \sum_{i=0}^{t-1}\mu^{s+t+1-i}(H^{i}{I,J}(M))+\mu^{s+t}(M)+\sum{i=t+1}^{s+t-1}\mu^{s+t-1-i}(H^{i}_{I,J}(M))$. As a consequence, it follows that if $I$ is a principal ideal of $R$ and $M$ is a minimax $R$-module, then $\mu^j(H^{i}_{I,J}(M))$ is finite for all $i\in\Bbb N_0$ and all $j\in\Bbb N_0$.