Upper bounds, cofiniteness, and artinianness of local cohomology modules defined by a pair of ideals
Let $R$ be a commutative noetherian ring, $I,J$ be two ideals of $R$, $M$ be an $R$-module, and $\mathcal{S}$ be a Serre class of $R$-modules. A positive answer to the Huneke$^,$s conjecture is given for a noetherian ring $R$ and minimax $R$-module $M$ of krull dimension less than 3, with respect to...
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Zusammenfassung: | Let $R$ be a commutative noetherian ring, $I,J$ be two ideals of $R$, $M$ be
an $R$-module, and $\mathcal{S}$ be a Serre class of $R$-modules. A positive
answer to the Huneke$^,$s conjecture is given for a noetherian ring $R$ and
minimax $R$-module $M$ of krull dimension less than 3, with respect to
$\mathcal{S}$. There are some results on cofiniteness and artinianness of local
cohomology modules with respect to a pair of ideals. For a ZD-module $M$ of
finite krull dimension and an integer $n\in\mathbb{N}$, if
$\lc^{i}_{I,J}(M)\in\mathcal{S}$ for all $i>n$, then
$\lc^{i}_{I,J}(M)/\fa^{j}\lc^{i}_{I,J}(M)\in\mathcal{S}$ for any
$\fa\in\tilde{W}(I,J)$, all $i\geq n$, and all $j\geq0$. By introducing the
concept of Seree cohomological dimension of $M$ with respect to $(I,J)$, for an
integer $r\in\mathbb{N}_0$, $\lc^{j}_{I,J}(R)\in\mathcal{S}$ for all $j>r$ iff
$\lc^{j}_{I,J}(M)\in\mathcal{S}$ for all $j>r$ and any finite $R$-module $M$. |
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DOI: | 10.48550/arxiv.1211.4204 |