arXiv subjects
Publications and source records attributed to Sh. Payrovi.
Let R be a Noetherian ring, I an ideal of R and M a ZD-module. Let S be a Melkersson subcategory with respect to I such that M/IM doesn't belong to S. We show that all maximal S-sequences on M in I, have equal length. If this common length is denoted by S-depth_I(M), then S-depth_I(M) = inf{i : H^i_I(M) doesn't belong to S}.
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 $μ^t(M)\leq\sum_{i=0}^{t}μ^{t-i}(H^{i}_{I,J}(M))$ and $μ^s(H^{t}_{I,J}(M))\leq \sum_{i=0}^{t-1}μ^{s+t+1-i}(H^{i}_{I,J}(M))+μ^{s+t}(M)+\sum_{i=t+1}^{s+t-1}μ^{s+t-1-i}(H^{i}_{I,J}(M))$. As a consequence, it follows that if $I$ is a principal ideal of $R$ and $M$ is a minimax $R$-module, then $μ^j(H^{i}_{I,J}(M))$ is finite for all $i\in\Bbb N_0$ and all $j\in\Bbb N_0$.
Let $R$ be a Noetherian ring, $I$ and $J$ two ideals of $R$ and $t$ an integer. Let $S$ be the class of Artinian $R$-modules, or the class of all $R$-modules $N$ with $\dim_RN\leq k$, where $k$ is an integer. It is proved that $\inf\{i: H^{i}_{I,J}(M)\notin S\}=\inf\{S-\depth_\frak{a}(M): \frak{a}\in \tilde{\rm W}(I,J)\}$, where $M$ is a finitely generated $R$-module, or is a $ZD$-module such that $M/\frak{a}M\notin S$ for all $\frak{a}\in \tilde{\rm W}(I,J)$. Let $\Supp_R H^{i}_{I,J}(M)$ be a finite subset of $\Max(R)$ for all $i<t$. It is shown that there are maximal ideals $\frak m_1, \frak m_2,\ldots,\frak m_k$ of $R$ such that $H^{i}_{I,J}(M)\cong H^{i}_{\frak m_1}(M)\oplus H^{i}_{\frak m_2}(M)\oplus\cdots\oplus H^{i}_{\frak m_k}(M)$ for all $i<t$.