Some results on local cohomology modules defined by a pair of ideals

ON GRADED LOCAL COHOMOLOGY MODULES DEFINED BY A PAIR OF IDEALS

Let $R = bigoplus_{n in mathbb{N}_{0}} R_{n}$ be a standardgraded ring, $M$ be a finitely generated graded $R$-module and $J$be a homogenous ideal of $R$. In this paper we study the gradedstructure of the $i$-th local cohomology module of $M$ defined by apair of ideals $(R_{+},J)$, i.e. $H^{i}_{R_{+},J}(M)$. Moreprecisely, we discuss finiteness property and vanishing of thegraded components $H^...

Serre Subcategories and Local Cohomology Modules with Respect to a Pair of Ideals

This paper is concerned with the relation between local cohomology modules defined by a pair of ideals and the Serre subcategories of the category of modules. We characterize the membership of local cohomology modules in a certain Serre subcategory from lower range or upper range.

Cofiniteness of Local Cohomlogy Based on a Non–Closed Support Defined by a Pair of Ideals

Some Results on Local Cohomology Modules

Let R be a commutative Noetherian ring, a an ideal of R, and let M be a finitely generated R-module. For a non-negative integer t, we prove that H a(M) is a-cofinite whenever H t a(M) is Artinian and H i a(M) is a-cofinite for all i < t. This result, in particular, characterizes the a-cofiniteness property of local cohomology modules of certain regular local rings. Also, we show that for a loca...

Local Cohomology Based on a Nonclosed Support Defined by a Pair of Ideals

We introduce an idea for generalization of a local cohomology module, which we call a local cohomology module with respect to a pair of ideals (I, J), and study their various properties. Some vanishing and nonvanishing theorems are given for this generalized version of local cohomology. We also discuss its connection with the ordinary local cohomology.