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

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

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

Generalized local cohomology and the Intersection Theorem

Let $R$ be commutative Noetherian ring and let \fa be an ideal of $R$. For complexes $X$ and $Y$ of $R$--modules we investigate the invariant \inf{\mathbf R}\Gamma_{\fa}({\mathbf R}\Hom_R(X,Y)) in certain cases. It is shown that, for bounded complexes $X$ and $Y$ with finite homology, \dim Y\le\dim{\mathbf R}\Hom_R(X,Y)\le\pd X+\dim(X\otimes^{\mathbf L}_RY)+\sup X which strengthen the Intersection Theorem. Here $\inf X$ and $\sup X$ denote the homological infimum, and supremum of the complex $X$, respectively.Comment: 13 page

Attached primes of the top local cohomology modules with respect to an ideal (II)

For a finitely generated module $M$, over a commutative Noetherian local ring $(R, \mathfrak{m})$, it is shown that there exist only a finite number of non--isomorphic top local cohomology modules $\mathrm{H}_{\mathfrak{a}}^{\mathrm{dim} (M)}(M)$, for all ideals $\mathfrak{a}$ of $R$. We present a reduced secondary representation for the top local cohomology modules with respect to an ideal. It is also shown that for a given integer $r\geq 0$, if $\mathrm{H}_{\mathfrak{a}}^{r}(R/\mathfrak {p})$ is zero for all $\mathfrak{p}$ in $\mathrm{Supp}(M)$, then $\mathrm{H}_{\mathfrak{a}}^{i}(M)= 0$ for all $i\geq r$.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

Cohen-Macaulay Loci of modules

The Cohen-Macaulay locus of any finite module over a noetherian local ring $A$ is studied and it is shown that it is a Zariski-open subset of \Spec A in certain cases. In this connection, the rings whose formal fibres over certain prime ideals are Cohen-Macaulay are studied.Comment: 18 pages, to appear in "Communications in Agebra