arXiv subjects

Kamran Divaani-Aazar

Publications and source records attributed to Kamran Divaani-Aazar.

51 records · Page 3Linked to original sources

arXiv2010

Finiteness properties of formal local cohomology modules and Cohen-Macaulayness

Let $\fa$ be an ideal of a local ring $(R,\fm)$ and $M$ a finitely generated $R$-module. We investigate the structure of the formal local cohomology modules ${\vpl}_nH^i_{\fm}(M/\fa^n M)$, $i\geq 0$. We prove several results concerning finiteness properties of formal local cohomology modules which indicate that these modules behave very similar to local cohomology modules. Among other things, we prove that if $\dim R\leq 2$ or either $\fa$ is principal or $\dim R/\fa\leq 1$, then $\Tor_j^R(R/\fa,{\vpl}_nH^i_{\fm}(M/\fa^n M))$ is Artinian for all $i$ and $j$. Also, we examine the notion $\fgrade(\fa,M)$, the formal grade of $M$ with respect to $\fa$ (i.e. the least integer $i$ such that ${\vpl}_nH^i_{\fm}(M/\fa^n M) \neq 0$). As applications, we establish a criterion for Cohen-Macaulayness of $M$, and also we provide an upper bound for cohomological dimension of $M$ with respect to $\fa$.

math.AC↗

arXiv2010

On the formal grade of finitely generated modules over local rings

Let \fa be an ideal of a local ring (R,\fm) and M a finitely generated R-module. This paper concerns the notion \fgrade(\fa,M), the formal grade of M with respect to \fa (i.e. the least integer i such that {\vpl}_nH^i_{\fm}(M/\fa^n M)\neq 0). We show that \fgrade(\fa,M)\geq \depth M-\cd_{\fa}(M), and as a result, we establish a new characterization of Cohen-Macaulay modules. As an application of this characterization, we show that if M is Cohen-Macaulay and L a pure submodule of M with the same support as M, then \fgrade(\fa,L)=\fgrade(\fa,M). Also, we give a generalization of the Hochster-Eagon result on Cohen-Macaulayness of invariant rings.

math.AC↗

arXiv2008

The finiteness dimension of local cohomology modules and its dual notion

Let \fa be an ideal of a commutative Noetherian ring R and M a finitely generated R-module. We explore the behavior of the two notions f_{\fa}(M), the finiteness dimension of M with respect to \fa, and, its dual notion q_{\fa}(M), the Artinianess dimension of M with respect to \fa. When (R,\fm) is local and r:=f_{\fa}(M) is less than f_{\fa}^{\fm}(M), the \fm-finiteness dimension of M relative to \fa, we prove that H^r_{\fa}(M) is not Artinian, and so the filter depth of \fa on M doesn't exceeds f_{\fa}(M). Also, we show that if M has finite dimension and H^i_{\fa}(M) is Artinian for all i>t, where t is a given positive integer, then H^t_{\fa}(M)/\fa H^t_{\fa}(M) is Artinian. It immediately implies that if q:=q_{\fa}(M)>0, then H^q_{\fa}(M) is not finitely generated, and so f_{\fa}(M)\leq q_{\fa}(M).

math.AC↗

arXiv2007

Generalized local cohomology and regularity of Ext modules

Let R be a Noetherian standard graded ring, and M and N two finitely generated graded R-modules. We introduce reg_R (M,N) by using the notion of generalized local cohomology instead of local cohomology, in the definition of regularity. We prove that reg_R (M,N)is finite in several cases. In the case that the base ring is a field, we show that reg_R (M,N)=reg (N)-indeg (M). This formula, together with a graded version of duality for generalized local cohomology, gives a formula for the minimum of the initial degrees of some Ext modules(in the case R is Cohen-Macaulay), of which the three usual definitions of regularity are special cases. Bounds for regularity of certain Ext modules are obtained, using the same circle of ideas.

math.AC↗

arXiv2007

A duality theorem for generalized local cohomology

We prove a duality theorem for graded algebras over a field that implies several known duality results : graded local duality, versions of Serre duality for local cohomology and of Suzuki duality for generalized local cohomology, and Herzog-Rahimi bigraded duality.

math.AC↗

arXiv2007

A criterion for rings which are locally valuation

Using the notion of cyclically pure injective modules, a characterization for rings which are locally valuation is established. As applications, new characterizations for Prufer domains and pure semi-simple rings are provided. Namely, we show that a domain R is Prufer if and only if two of the three classes of pure injective, cyclically pure injective and RD-injective modules are equal. Also, we prove that a commutative ring R is pure semi-simple if and only if every R-module is cyclically pure injective.

math.AC↗

arXiv2007

Vanishing of the top local cohomology modules over Noetherian rings

Let R be a (not necessarily local) Noetherian ring and M a finitely generated R-module of finite dimension d. Let \fa be an ideal of R and \fM denote the intersection of all prime ideals \fp in Supp_RH^d_{\fa}(M). It is shown that H^d_{\fa}(M)\simeq H^d_{\fM}(M)/\displaystyle{\sum_{n\in \mathbb{N}}}<\fM>(0:_{H^d_{\fM}(M)}\fa^n), where for an Artinian R-module A we put <\fM>A=\cap_{n\in \mathbb{N}} \fM^nA. As a consequence, it is proved that for all ideals \fa of R, there are only finitely many non-isomorphic top local cohomology modules H^d_{\fa}(M) having the same support. In addition, we establish an analogue of the Lichtenbaum-Hartshorne Vanishing Theorem over rings that need not be local.

math.AC↗

arXiv2005

Some criteria of cyclically pure injective modules

The structure of cyclically pure injective modules over a commutative ring $R$ is investigated and several characterizations for them are presented. In particular, we prove that a module $D$ is cyclically pure injective if and only if $D$ is isomorphic to a direct summand of a module of the form $\Hom_R(L,E)$ where $L$ is the direct sum of a family of finitely presented cyclic modules and $E$ is an injective module. Also, we prove that over a quasi-complete Noetherian ring $(R,\fm)$ an $R$-module $D$ is cyclically pure injective if and only if there is a family $\{C_λ\}_{λ\in Λ}$ of cocyclic modules such that $D$ is isomorphic to a direct summand of $Π_{λ\in Λ}C_λ$. Finally, we show that over a complete local ring every finitely generated module which has small cofinite irreducibles is cyclically pure injective.

math.AC↗

arXiv2005

Associated primes of local cohomology modules of weakly Laskerian modules

The notion of weakly Laskerian modules was introduced recently by the authors. Let $R$ be a commutative Noetherian ring with identity, $\fa$ an ideal of $R$, and $M$ a weakly Laskerian module. It is shown that if $\fa$ is principal, then the set of associated primes of the local cohomology module $H_{\fa}^i(M)$ is finite for all $i\geq 0$. We also prove that when $R$ is local, then $\Ass_R(H_{\fa}^i(M))$ is finite for all $i\geq 0$ in the following cases: (1) $\dim R\leq 3$, (2) $\dim R/\fa\leq 1$, (3) $M$ is Cohen-Macaulay and for any ideal $\fb$, with $l=\grade(\fb,M)$, $\Hom_R(R/\fb,H_{\fb}^{l+1}(M))$ is weakly Laskerian.

math.AC↗

arXiv2005

Localization of tight closure in two-dimensional rings

It is shown that tight closure commutes with localization in any two dimensional ring $R$ of prime characteristic if either $R$ is a Nagata ring or $R$ possesses a weak test element. Moreover, it is proved that tight closure commutes with localization at height one prime ideals in any ring of prime characteristic.

math.AC↗

arXiv2004

Artinianness of local cohomology modules of ZD-modules

This paper centers around Artinianness of the local cohomology of $ZD$-modules. Let $\fa$ be an ideal of a commutative Noetherian ring $R$. The notion of $\fa$-relative Goldie dimension of an $R$-module $M$, as a generalization of that of Goldie dimension is presented. Let $M$ be a $ZD$-module such that $\fa$-relative Goldie dimension of any quotient of $M$ is finite. It is shown that if $\dim R/\fa=0$, then the local cohomology modules $H^i_{\fa}(M)$ are Artinian. Also, it is proved that if $d=\dim M$ is finite, then $H^d_{\fa}(M)$ is Artinian, for any ideal $\fa$ of $R$ . These results extend the previously known results concerning Artinianness of local cohomology of finitely generated modules.

math.AC↗

arXiv2004

Associated primes of local cohomology module

Let $\fa$ be an ideal of a commutative Noetherian ring $R$ and $M$ a finitely generated $R$-module. Let $t$ be a natural integer. It is shown that there is a finite subset $X$ of $\Spec R$, such that $\Ass_R(H_{\fa}^t(M))$ is contained in $X$ union with the union of the sets $\Ass_R(\Ext_R^j(R/\fa,H_{\fa}^i(M)))$, where $0\leq i<t$ and $0\leq j\leq t^2+1$. As an immediate consequence, we deduce that the first non $\fa$-cofinite local cohomology module of $M$ with respect to $\fa$ has only finitely many associated prime ideals.

math.AC↗

arXiv2004

Cofiniteness of generalized local cohomology modules

Let $\fa$ denote an ideal of a commutative Noetherian ring $R$ and $M$ and $N$ two finitely generated $R$-modules with $\pd M< \infty$. It is shown that if $\fa$ is principal or $R$ is complete local and $\fa$ a prime ideal with $\dim R/\fa=1$, then the generalized local cohomology module $H^i_{\fa}(M,N)$ is $\fa$-cofinite for all $i \geq 0$. This provides an affirmative answer for the above ideal $\fa$ to a question proposed in [{\bf 13}].

math.AC↗

arXiv2004

Associated prime submodules of finitely generated modules

Let $R$ be a commutative ring with identity. For a finitely generated $R$-module $M$, the notion of associated prime submodules of $M$ is defined. It is shown that this notion inherits most of essential properties of the usual notion of associated prime ideals. In particular, it is proved that for a Noetherian multiplication module $M$, the set of associated prime submodules of $M$ coincides with the set of $M$-radicals of primary submodules of $M$ which appear in a minimal primary decomposition of the zero submodule of $M$. Also, Anderson's theorem [{\bf 2}] is extended to minimal prime submodules in a certain type of modules.

math.AC↗

arXiv2004

A new characterization of commutative Artinian rings

Let $R$ be a commutative Noetherian ring. It is shown that $R$ is Artinian if and only if every $R$-module is good, if and only if every $R$-module is representable. As a result, it follows that every nonzero submodule of any representable $R$-module is representable if and only if $R$ is Artinian. This provides an answer to a question which is investigated in [{\bf 1}].

math.AC↗