Depth of an ideal on ZD-modules

Let R be a Noetherian ring, I an ideal of R and M a ZD-module. Let S be a Serre subcategory of the category of R-modules satisfying the condition CI, and let I contain a maximal S-sequence on M. We show that all maximal S-sequences on M in I, have the same length. If this common length is denoted by S-depth(I,M), then S-depth(I,M) = inf{i:ExtiR(R/I,M) ?S} = inf{i:HiI (M)?S}. Also some properties of this notion are investigated. It is proved that S-depth(I,M) = inf{depthMp : p ?V (I) and R/p ? S} = inf{S-depth(p,M) : p ? V (I) and R/p ? S} whenever S is a Serre subcategory closed under taking injective hulls, and M is a ZD-module.

Lower Bounds for Local Cohomology Modules with Respect to a Pair of Ideals

Let R be a Noetherian ring, I and J two ideals of R, M an R-module and t an integer. Let S be a Serre subcategory of the category of R-modules satisfying the condition CI, and N be a finitely generated R-module with Supp RN= V (𝔞) for some [Formula: see text]. It is shown that if [Formula: see text] for all i < t and all j < t-i, then [Formula: see text] for all i < t. Let S be the class of all R-modules N with dim R N ≤ k, where k is an integer. It is proved that if [Formula: see text] for all i < t and all [Formula: see text], then [Formula: see text] for all i < t. It follows that [Formula: see text].

Filter Depth and Cofiniteness of Local Cohomology Modules Defined by a Pair of Ideals

Let I, J be ideals of a commutative Noetherian local ring (R, 𝔪) and let M be a finite R-module. The f-depth of M with respect to I is the least integer r such that [Formula: see text] is not Artinian. In this paper we show that [Formula: see text] is the least integer such that the local cohomology module with respect to a pair of ideals I, J is not Artinian. As a consequence, it follows that [Formula: see text] is (I,J)-cofinite for all [Formula: see text]. In addition, we show that for a Serre subcategory 𝖲, if [Formula: see text] belongs to 𝖲 for all i > n and if 𝔟 is an ideal of R such that [Formula: see text] belongs to 𝖲, then the module [Formula: see text] belongs to 𝖲.

Local Cohomology Modules, Serre Subcategories and Derived Functors of Torsion Functors

In this research, by using filter regular sequences, we obtain some exact sequences of right or left derived functors of local cohomology modules. Then we use them to gain some conditions under which a right or left derived functor of some special functors over local cohomology modules belongs to a Serre subcategory. These results can conclude some generalizations of previous results in this context or regain some of them.

LOCAL-GLOBAL PRINCIPLE FOR THE FINITENESS AND ARTINIANNESS OF GENERALISED LOCAL COHOMOLOGY MODULES

Let $\mathcal S$ be a Serre subcategory of the category of $R$-modules, where $R$ is a commutative Noetherian ring. Let $\mathfrak a$ and $\mathfrak b$ be ideals of $R$ and let $M$ and $N$ be finite $R$-modules. We prove that if $N$ and $H^i_{\mathfrak a}(M,N)$ belong to $\mathcal S$ for all $i\lt n$ and if $n\leq \mathrm {f}$-$\mathrm {grad}({\mathfrak a},{\mathfrak b},N )$, then $\mathrm {Hom}_{R}(R/{\mathfrak b},H^n_{{\mathfrak a}}(M,N))\in \mathcal S$. We deduce that if either $H^i_{\mathfrak a}(M,N)$ is finite or $\mathrm {Supp}\,H^i_{\mathfrak a}(M,N)$ is finite for all $i\lt n$, then $\mathrm {Ass}\,H^n_{\mathfrak a}(M,N)$ is finite. Next we give an affirmative answer, in certain cases, to the following question. If, for each prime ideal ${\mathfrak {p}}$ of $R$, there exists an integer $n_{\mathfrak {p}}$ such that $\mathfrak b^{n_{\mathfrak {p}}} H^i_{\mathfrak a R_{\mathfrak {p}}}({M_{\mathfrak {p}}},{N_{\mathfrak {p}}})=0$ for every $i$ less than a fixed integer $t$, then does there exist an integer $n$ such that $\mathfrak b^nH^i_{\mathfrak a}(M,N)=0$ for all $i\lt t$? A formulation of this question is referred to as the local-global principle for the annihilation of generalised local cohomology modules. Finally, we prove that there are local-global principles for the finiteness and Artinianness of generalised local cohomology modules.

Towards a theory of R-modules modulo a Serre category

This note is a brief excursion into a new theory of modules over a commutative ring R modulo a Serre subcategory S of the category of R-modules, in the sense that the modules of S are regarded as trivial. As a demonstration of the theory we have chosen to extend the primary decomposition theorem for submodules of a Noetherian .R-module from the familiar case of S trivial (i.e. the only R-module of S is the zero module) to the general case.