Asymptotic behavior of $j$-multiplicities

Let $R= \oplus_{n\in \mathbb{N}_0}R_n$ be a Noetherian homogeneous ring with local base ring $(R_0,\mathfrak{m}_0)$. Let $R_+= \oplus_{n\in \mathbb{N}}R_n$ denote the irrelevant ideal of $R$ and let $M=\oplus_{n\in \mathbb{Z}}M_n$ be a finitely generated graded $R$-module. When $\dim(R_0)\leq 2$ and $\mathfrak{q}_0$ is an arbitrary ideal of $R_0$, we show that the $j$-multiplicity of the graded local cohomology module $j_0({\mathfrak{q}_0},H_{R_+}^i(M)_n)$ has a polynomial behavior for all $n\ll0$.

Notes on endomorphisms, local cohomology and completion

Let M M denote a finitely generated module over a Noetherian ring R R . For an ideal I ⊂ R I \subset R there is a study of the endomorphisms of the local cohomology module H I g ( M ) , g = g r a d e ( I , M ) , H^g_I(M), g = grade(I,M), and related results. Another subject is the study of left derived functors of the I I -adic completion Λ i I ( H I g ( M ) ) \Lambda ^I_i(H^g_I(M)) , motivated by a characterization of Gorenstein rings given in [25]. This provides another Cohen-Macaulay criterion. The results are illustrated by several examples. There is also an extension to the case of homomorphisms of two different local cohomology modules.

Attached primes of local cohomology modules with respect to a pair of ideals

The paper shows an extension of the Lichtenbaum–Hartshorne Vanishing theorem for local cohomology modules with respect to a pair of ideals. We also study the attached primes of top local cohomology module [Formula: see text] where [Formula: see text] In the case, where [Formula: see text] we show that [Formula: see text]

On the cominimaxness of generalized local cohomology modules

The local cohomology theory plays an important role in commutative algebra and algebraic geometry. The I-cofiniteness of local cohomology modules is one of interesting properties which has been studied by many mathematicians. The I-cominimax modules is an extension of I-cofinite modules which was introduced by Hartshorne. An R-module M is I-cominimax if Supp_R(M)\subseteq V(I) and Ext^i_R(R/I,M) is minimax for all i\ge 0. In this paper, we show some conditions such that the generalized local cohomology module H^i_I(M,N) is I-cominimax for all i\ge 0. We show that if H^i_I(M,K) is I-cofinite for all i<t and all finitely generated R-module K, then H^i_I(M,N) is I-cominimax for all i<t and all minimax R-module N. If M is a finitely generated R-module, N is a minimax R-module and t is a non-negative integer such that dim Supp_R(H^i_I(M,N))\le 1 for all i<t then H^i_I(M,N) is I-cominimax for all i<t. When dim R/I\le 1 and H^i_I(N) is I-cominimax for all i\ge 0 then H^i_I(M,N) is I-cominimax for all i\ge 0.

Cominimaxness with respect to ideals of dimension two and local cohomology

Let [Formula: see text] be a commutative Noetherian ring, [Formula: see text] an ideal of [Formula: see text] and [Formula: see text] an [Formula: see text]-module with [Formula: see text]. We get equivalent conditions for top local cohomology module [Formula: see text] to be Artinian and [Formula: see text]-cofinite Artinian separately. In addition, we prove that if [Formula: see text] is a local ring such that [Formula: see text] is minimax, for each [Formula: see text], then [Formula: see text] is minimax [Formula: see text]-module for each [Formula: see text] and for each finitely generated [Formula: see text]-module [Formula: see text] with [Formula: see text] and [Formula: see text]. As a consequence we prove that if [Formula: see text] and [Formula: see text], then [Formula: see text] is [Formula: see text]-cominimax if (and only if) [Formula: see text], [Formula: see text] and [Formula: see text] are minimax. We also prove that if [Formula: see text] and [Formula: see text] such that [Formula: see text] is minimax for all [Formula: see text], then [Formula: see text] is [Formula: see text]-cominimax for all [Formula: see text] if (and only if) [Formula: see text] is minimax for all [Formula: see text].

Modules with minimax Cousin cohomologies

Let R be a commutative Noetherian ring with non-zero identity and let X be an arbitrary R-module. In this paper, we show that if all the cohomology modules of the Cousin complex for X are minimax, then the following hold for any prime ideal p of R and for every integer n less than X, the height of p: (i) the nth Bass number of X with respect to p is finite; (ii) the nth local cohomology module of Xp with respect to pRp is Artinian.

General formal local cohomology modules

Let (R,m) be a local ring, Φ a system of ideals of R and M a finitely generated R-module. In this paper, we define and study general formal local cohomology modules. We denote the ith general formal local cohomology module M with respect to Φ by FiΦ(M) and we investigate some finiteness and Artinianness properties of general formal local cohomology modules.

Comparison of local cohomlogy modules of a ring and its amalgamated algebra along a given ideal

Let [Formula: see text] be a commutative Noetherian ring and let [Formula: see text] and [Formula: see text] be ideals of [Formula: see text]. The main purpose of this paper is to compare of the finiteness properties of [Formula: see text] and [Formula: see text], where [Formula: see text] is the [Formula: see text]th local cohomology module functor with respect to [Formula: see text] and [Formula: see text] is the amalgamated of [Formula: see text] along [Formula: see text].

Hilbert–Samuel multiplicity and Northcott’s inequality relative to an Artinian module

Let [Formula: see text] be a commutative quasi-local ring (with identity [Formula: see text]), and let [Formula: see text] be an [Formula: see text]-ideal such that [Formula: see text]. For [Formula: see text] an Artinian [Formula: see text]-module of N-dimension [Formula: see text], we introduce the notion of Hilbert-coefficients of [Formula: see text] relative to [Formula: see text] and give several properties. When [Formula: see text] is a co-Cohen–Macaulay [Formula: see text]-module, we establish the Northcott’s inequality for Artinian modules. As applications, we show some formulas involving the Hilbert coefficients and we investigate the behavior of these multiplicities when the module is the local cohomology module.