A study of Cousin complexes through the dualizing complexes

For the Cousin complex of certain modules, we investigate finiteness of cohomology modules, local duality property and injectivity of its terms. The existence of canonical modules of Noetherian non-local rings and the Cousin complexes of them with respect to the height filtration are discussed .Comment: 15 pages. Accepted for publication in Communication Algebr

Associated primes and cofiniteness of local cohomology modules

Let $\mathfrak{a}$ be an ideal of Noetherian ring $R$ and let $M$ be an $R$-module such that $\mathrm{Ext}^i_R(R/\mathfrak{a},M)$ is finite $R$-module for every $i$. If $s$ is the first integer such that the local cohomology module $\mathrm{H}^s_\mathfrak{a}(M)$ is non $\mathfrak{a}$-cofinite, then we show that $\mathrm{Hom}_{R}(R/\mathfrak{a}, \mathrm{H}^s_\mathfrak{a}(M))$ is finite. Specially, the set of associated primes of $\mathrm{H}^s_\mathfrak{a}(M)$ is finite. Next assume $(R,\mathfrak{m})$ is a local Noetherian ring and $M$ is a finitely generated module. We study the last integer $n$ such that the local cohomology module $\mathrm{H}^n_\mathfrak{a}(M)$ is not $\mathfrak{m}$-cofinite and show that $n$ just depends on the support of $M$.Comment: 9 page

Cohomological dimension of complexes

In the derived category of the category of modules over a commutative Noetherian ring $R$, we define, for an ideal \fa of $R$, two different types of cohomological dimensions of a complex $X$ in a certain subcategory of the derived category, namely \cd(\fa, X)=\sup\{\cd(\fa, \H_{\ell}(X))-\ell|\ell\in\Bbb Z\} and -\inf{\mathbf R}\G_{\fa}(X), where \cd(\fa, M)=\sup\{\ell\in\Bbb Z|\H^{\ell}_{\fa}(M)\neq 0\} for an $R$--module $M$. In this paper, it is shown, among other things, that, for any complex $X$ bounded to the left, -\inf {\mathbf R}\G_{\fa}(X)\le\cd(\fa, X) and equality holds if indeed $\H(X)$ is finitely generated.Comment: 13 page

Finiteness of extension functors of local cohomology modules

Let $R$ be a commutative Noetherian ring, \fa an ideal of $R$ and $M$ a finitely generated $R$--module. Let $t$ be a non-negative integer such that \H^i_\fa(M) is \fa--cofinite for all $i<t$. It is well--known that \Hom_R(R/\fa,\H^t_\fa(M)) is finitely generated $R$--module. In this paper we study the finiteness of \Ext^1_R(R/\fa,\H^t_\fa(M)) and \Ext^2_R(R/\fa,\H^t_\fa(M)).Comment: 5 page

Complexes of C-projective modules

Inspired by a recent work of Buchweitz and Flenner, we show that, for a semidualizing bimodule $C$, $C$--perfect complexes have the ability to detect when a ring is strongly regular. It is shown that there exists a class of modules which admit minimal resolutions of $C$--projective modules.Comment: 10 pages, To appear in Bulletin of the Iranian Mathematical Societ

Linkage of modules with respect to a semidualizing module

The notion of linkage with respect to a semidualizing module is introduced. It is shown that over a Cohen-Macaulay local ring with canonical module, every Cohen-Macaulay module of finite Gorenstein injective dimension is linked with respect to the canonical module. For a linked module $M$ with respect to a semidualizing module, the connection between the Serre condition $(S_n)$ on $M$ with the vanishing of certain local cohomology modules of its linked module is discussed.Comment: 16 pages. arXiv admin note: text overlap with arXiv:1507.00036, arXiv:1407.654

Some characterizations of special rings by delta-invariant

This paper is devoted to present some characterizations for a local ring to be generically Gorenstein and Gorenstein by means of $\delta$-invariant and linkage theory.Comment: 12 pages, minor changes in the title and abstract; typos correcte

Sequence of Exact Zero-Divisors

The concept of a sequence of exact zero-divisors on a noetherian local ring is defined and studied. Some properties of sequences of exact zero-divisors are compared with regular sequences.Comment: 18 pages, with some correction

Linkage of modules and the Serre conditions

Let $R$ be semiperfect commutative Noetherian ring and $C$ be a semidualizing $R$--module. The connection of the Serre condition $(S_n)$ on a horizontally linked $R$-module of finite \gc-dimension with the vanishing of certain cohomology modules of its linked module is discussed. As a consequence, it is shown that under some conditions Cohen-Macaulayness is preserved under horizontally linkage.Comment: 21 pages, final version will appear in Journal of Pure and Applied Algebr

A study of strongly Cohen-Macaulay ideals by delta invariant

Let $R$ be a Cohen-Macaulay local ring, $I$ a strongly Cohen-Macaulay ideal of $R$. We show that $R/{\rm Ann}_R(I)$ is a maximal Cohen-Macaulay $R$-module by means of the delta-invariant. Also it is shown that there exists a Cohen-Macaulay ideal $J$ of $R$ such that the $\delta_{R/J}$-invariant of all Koszul homologies of $R$ with respect $I$ is zero.Comment: 7 pages; final version; to appear in Bulletin of the Iranian Mathematical Societ