Commutative ring theory: Regular rings

Finiteness properties of local cohomology modules(局所コホモロジー加群の有限性について)

Let R be a commutative Noetherian ring, I and J be two ideals of R and M be an R-module (not necessary I-torsion). In this paper among other things, it is shown that if dim M ≤ 1, then the R-module ExtRi(R/I,M) is finitely generated, for all i ≥ 0, if and only if the R-module ExtRi(R/I,M) is finitely generated, for i = 0, 1. As a consequence, we prove that if M is finitely generated and t ∈ ℕ such that the R-module HI,Ji(M) is FD≤1 (or weakly Laskerian) for all i < t, then HI,Ji(M) is (I,J)-cofinite for all i < t and for any FD≤0 (or minimax) submodule N of HI,Jt(M), the R-modules HomR(R/I,HI,Jt(M)/N) and ExtR1(R/I,H I,Jt(M)/N) are finitely generated. Also it is shown that if dim M/&#x1d51e;M ≤ 1 (e.g. dim R/&#x1d51e; ≤ 1) for all &#x1d51e; ∈W(I,J), then the local cohomology module HI,Ji(M) is (I,J)-cofinite for all i ≥ 0.

Cofiniteness of local cohomology modules for a pair of ideals for small dimensions

Let $R$ be a commutative Noetherian ring. The notion of regular sequences with respect to a Serre class of $R$-modules is introduced and some of their essential properties are given. Then in the local case, we explore a theory of Cohen–Macaulayness with respect to Serre classes.

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Cohen-Macaulayness with respect to Serre classes

‎We introduce a generalization of the notion of‎ depth of an ideal on a module by applying the concept of‎ local cohomology modules with respect to a pair‎ ‎of ideals‎. ‎We also introduce the concept of $(I,J)$-Cohen--Macaulay modules as a generalization of concept of Cohen--Macaulay modules‎. ‎These kind of modules are different from Cohen--Macaulay modules‎, as an example shows‎. ‎Also an artinian result for such modules is given‎.

http://jas.shahroodut.ac.ir/article_482_3406cd1fa845d38b77f2556344be6005.pdf

The concept of (i; j)-cohen macaulay modules

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 ${{\mathrm{\mathrm{H}}}}^{i}_{I,J}(M)\in {\mathcal {S}}$ for all $i>n$, then ${{\mathrm{\mathrm{H}}}}^{i}_{I,J}(M)/{\mathfrak {a}}^{j}{{\mathrm{\mathrm{H}}}}^{i}_{I,J}(M)\in {\mathcal {S}}$ for any ${\mathfrak {a}}\in \tilde{W}(I,J)$, all $i\ge n$, and all $j\ge 0$. By introducing the concept of Serre cohomological dimension of M with respect to (I, J), for an integer $r\in {\mathbb {N}}_0$, ${{\mathrm{\mathrm{H}}}}^{j}_{I,J}(R)\in {\mathcal {S}}$ for all $j>r$ iff ${{\mathrm{\mathrm{H}}}}^{j}_{I,J}(M)\in {\mathcal {S}}$ for all $j>r$ and any finite R-module M.

Cofiniteness and Artinianness of certain local cohomology modules

Let $(R,\fm)$ be a local ring and $I$ an ideal. The aim of the present paper is twofold. At first we continue the investigation to compare $\fgrade(I,R)$ with $\depth R/I$ and further we derive some results on the vanishing of Lyubeznik numbers.

On the relation between formal grade and depth with a view toward vanishing of Lyubeznik numbers

Some affirmative answers are given to Huneke's problems. The calculation of local cohomology modules with respect to an arbitrary pair of ideals $I,J$ can be reduced to calculation of local cohomology modules with respect to a pair of ideala whose first ideal is generated by any $k$-regular sequence in $I$.

Some results on the local cohomology modules with respect to 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 $\mathcal{S}$. 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 $M$.

Upper bounds, cofiniteness, and artinianness of local cohomology modules defined by a pair of ideals

Let (R,&#x1d52a;) be a local ring and I an ideal. The aim of the present paper is twofold. At first we continue the investigation to compare fgrade(I,R) with depth R∕I and further we derive some results on the vanishing of Lyubeznik numbers.

Kh. Ahmadi-Amoli

Papers

Cofiniteness and Artinianness of certain local cohomology modules

On the relation between formal grade and depth with a view toward vanishing of Lyubeznik numbers

Some results on the local cohomology modules with respect to a pair of ideals

Upper bounds, cofiniteness, and artinianness of local cohomology modules defined by a pair of ideals

On the relation between formal grade and depth with a view toward vanishing of Lyubeznik numbers