arXiv subjects

Kamran Divaani-Aazar

Publications and source records attributed to Kamran Divaani-Aazar.

At least 37 records · Page 2Linked to original sources

arXiv2018

Local Homology, Koszul Homology and Serre Classes

Given a Serre class $\mathcal{S}$ of modules, we compare the containment of the Koszul homology, Ext modules, Tor modules, local homology, and local cohomology in $\mathcal{S}$ up to a given bound $s \geq 0$. As some applications, we give a full characterization of noetherian local homology modules. Further, we establish a comprehensive vanishing result which readily leads to the formerly known descriptions of the numerical invariants width and depth in terms of Koszul homology, local homology, and local cohomology. Also, we immediately recover a few renowned vanishing criteria scattered about the literature.

math.AC↗

arXiv2018

Stable Under Specialization Sets and Cofiniteness

Let $R$ be a commutative noetherian ring, and $\mathcal{Z}$ a stable under specialization subset of $\Spec(R)$. We introduce a notion of $\mathcal{Z}$-cofiniteness and study its main properties. In the case $\dim(\mathcal{Z})\leq 1$, or $\dim(R)\leq 2$, or $R$ is semilocal with $\cd(\mathcal{Z},R) \leq 1$, we show that the category of $\mathcal{Z}$-cofinite $R$-modules is abelian. Also, in each of these cases, we prove that the local cohomology module $H^{i}_{\mathcal{Z}}(X)$ is $\mathcal{Z}$-cofinite for every homologically left-bounded $R$-complex $X$ whose homology modules are finitely generated and every $i \in \mathbb{Z}$.

math.AC↗

arXiv2017

Local homology, finiteness of Tor modules and cofiniteness

Let $\frak a$ be an ideal of a commutative noetherian ring $R$ with unity and $M$ an $R$-module supported at $\V(\fa)$. Let $n$ be the supermum of the integers $i$ for which $H^{\fa}_i(M)\neq 0$. We show that $M$ is $\fa$-cofinite if and only if the $R$-module $\Tor^R_i(R/\fa,M)$ is finitely generated for every $0\leq i\leq n$. This provides a hands-on and computable finitely-many-steps criterion to examine $\mathfrak{a}$-confiniteness. Our approach relies heavily on the theory of local homology which demonstrates the effectiveness and indispensability of this tool.

math.AC↗

arXiv2016

Weakly Laskerian rings versus Noetherian rings

Let R be a commutative ring with identity. We investigate some ring-theoretic properties of weakly Laskerian R-modules. Our results indicate that weakly Laskerian rings behave as Noetherian ones in many respects. However, we provide some examples to illustrate the strange behavior of these rings in some other respects.

math.AC↗

arXiv2015

A criterion for dualizing modules

We establish a characterization of dualizing modules among semidualizing modules. Let R be a finite dimensional commutative Noetherian ring with identity and C a semidualizing R-module. We show that C is a dualizing R-module if and only if Tor_i^R(E,E') is C- injective for all C-injective R-modules E and E' and all i\geq 0.

math.AC↗

arXiv2014

Gorenstein homology, relative pure homology and virtually Gorenstein rings

We consider the following question: Is Gorenstein homology a X-pure homology, in the sense defined by Warfield, for a class X of modules? Let GP denote the class of Gorenstein projective modules. We prove that over a commutative Noetherian ring R of finite Krull dimension, Gorenstein homology is a GP-pure homology if and only if R is virtually Gorenstein.

math.AC↗

arXiv2013

Filter-regular sequences, almost complete intersections and Stanley's conjecture

Let $K$ be a field and $I$ a monomial ideal of the polynomial ring $S=K[x_1,..., x_n]$ generated by monomials $u_1,u_2,..., u_t$. We show that $S/I$ is pretty clean if either: 1) $u_1,u_2,..., u_t$ is a filter-regular sequence, 2) $u_1,u_2,..., u_t$ is a $d$-sequence; or 3) $I$ is almost complete intersection. In particular, in each of these cases, $S/I$ is sequentially Cohen-Macaulay and both Stanley's and $h$-regularity conjectures, on Stanley decompositions, hold for $S/I$. Also, we prove that if $I$ is the Stanley-Reisner ideal of a locally complete intersection simplicial complex on $[n]$, then Stanley's conjecture holds for $S/I$.

math.AC↗

arXiv2013

Almost complete intersections and Stanley's conjecture

Let $K$ be a field and $I$ a monomial ideal of the polynomial ring $S=K[x_1,\ldots, x_n]$. We show that if either: 1) $I$ is almost complete intersection, 2) $I$ can be generated by less than four monomials; or 3) $I$ is the Stanley-Reisner ideal of a locally complete intersection simplicial complex on $[n]$, then Stanley's conjecture holds for $S/I$.

math.AC↗

arXiv2013

Pretty cleanness and filter-regular sequences

Let $K$ be a field and $S=K[x_1,\ldots, x_n]$. Let $I$ be a monomial ideal of $S$ and $u_1,\ldots, u_r$ be monomials in $S$ which form a filter-regular sequence on $S/I$. We show that $S/I$ is pretty clean if and only if $S/(I,u_1,\ldots, u_r)$ is pretty clean.

math.AC↗

arXiv2013

Gorenstein injective envelopes of Artinian modules

Let R be a commutative Noetherian ring and A an Artinian R-module. We prove that if A has finite Gorenstein injective dimension, then A possesses a Gorenstein injective envelope which is special and Artinian. This, in particular, yields that over a Gorenstein ring any Artinian module possesses a Gorenstein injective envelope which is special and Artinian.

math.AC↗

arXiv2013

The Existence of Relative pure Injective Envelopes

Let $\mathcal{S}$ be a class of finitely presented $R$-modules such that $R\in \mathcal{S}$ and $\mathcal{S}$ has a subset $\mathcal{S}^*,$ with the property that for any $U\in \mathcal{S}$ there is a $U^*\in \mathcal{S}^*$ with $U^*\cong U.$ We show that the class of $\mathcal{S}$-pure injective $R$-modules is preenveloping. As an application, we deduce that the left global $\mathcal{S}$-pure projective dimension of $R$ is equal to its left global $\mathcal{S}$-pure injective dimension. As our main result, we prove that, in fact, the class of $\mathcal{S}$-pure injective $R$-modules is enveloping.

math.AC↗

arXiv2012

On the existence of certain modules of finite Gorenstein homological dimensions

Let (R,m) be a commutative Noetherian local ring. It is known that R is Cohen-Macaulay if there exists either a nonzero finitely generated R-module of finite injective dimension or a nonzero Cohen-Macaulay R-module of finite projective dimension. In this paper, we investigate the Gorenstein analogues of these facts.

math.AC↗

arXiv2012

On the vanishing of local homology modules

Let $R$ be a commutative Noetherian ring, $\fa$ an ideal of $R$ and $M$ an $R$-module. We intend to establish the dual of Grothendieck's Vanishing Theorem for local homology modules. We conjecture that $H^{\fa}_i(M)=0$ for all $i>\Mag_RM$. We prove this in several cases.

math.AC↗

arXiv2012

The derived category analogue of the Hartshorne-Lichtenbaum Vanishing Theorem

Let $\fa$ be an ideal of a local ring $(R,\fm)$ and $X$ a $d$-dimensional homologically bounded complex of $R$-modules whose all homology modules are finitely generated. We show that $H^d_{\fa}(X)=0$ if and only if $\dim \hat{R}/\fa \hat{R}+\fp>0$ for all prime ideals $\fp$ of $\hat{R}$ such that $\dim \hat{R}/\fp-\inf (X\otimes_R\hat{R})_{\fp}=d$.

math.AC↗

arXiv2012

Local homology and Gorenstein flat modules

Let $R$ be a commutative Noetherian ring, $\fa$ an ideal of $R$ and $\mathcal{D}(R)$ denote the derived category of $R$-modules. We investigate the theory of local homology in conjunction with Gorenstein flat modules. Let $X$ be a homologically bounded to the right complex and $Q$ a bounded to the right complex of Gorenstein flat $R$-modules such that $Q$ and $X$ are isomorphic in $\mathcal{D}(R)$. We establish a natural isomorphism ${\bf L}Λ^{\fa}(X)\simeq Λ^{\fa}(Q)$ in $\mathcal{D}(R)$ which immediately asserts that $\sup {\bf L}Λ^{\fa}(X)\leq \Gfd_RX$. This isomorphism yields several consequences. For instance, in the case $R$ possesses a dualizing complex, we show that $\Gfd_R {\bf L}Λ^{\fa}(X)\leq \Gfd_RX$. Also, we establish a criterion for regularity of Gorenstein local rings.

math.AC↗

arXiv2012

Foxby equivalence, local duality and Gorenstein homological dimensions

Let $(R,\fm)$ be a local ring and $(-)^{\vee}$ denote the Matlis duality functor. We investigate the relationship between Foxby equivalence and local duality through generalized local cohomology modules. Assume that $R$ possesses a normalized dualizing complex $D$ and $X$ and $Y$ are two homologically bounded complexes of $R$-modules with finitely generated homology modules. We present several duality results for $\fm$-section complex ${\bf R}Γ_{\fm}({\bf R}\Hom_R(X,Y))$. In particular, if G-dimension of $X$ and injective dimension of $Y$ are finite, then we show that $${\bf R}Γ_{\fm}({\bf R}\Hom_R(X,Y))\simeq ({\bf R}\Hom_R(Y,D\otimes_ R^{\bf L}X))^{\vee}.$$ We deduce several applications of these duality results. In particular, we establish Grothendieck's non-vanishing Theorem in the context of generalized local cohomology modules.

math.AC↗

arXiv2011

A unified approach to formal local cohomology and local Tate cohomology

Let R be a commutative Noetherian ring. We introduce a theory of formal local cohomology for complexes of R-modules. As an application, we establish some relations between formal local cohomology, local homology, local cohomology and local Tate cohomology through some natural isomorphisms. We investigate vanishing of formal local cohomology modules. Also, we give a characterization of Cohen-Macaulay complexes.

math.AC↗

arXiv2010

Generalized local cohomology modules and homological Gorenstein dimensions

Let \fa be an ideal of a commutative Noetherian ring R and M and N two finitely generated R-modules. Let \cd_{\fa}(M,N) denote the supremum of the i's such that H^i_{\fa}(M,N)\neq 0. First, by using the theory of Gorenstein homological dimensions, we obtain several upper bounds for \cd_{\fa}(M,N). Next, over a Cohen-Macaulay local ring (R,\fm), we show that \cd_{\fm}(M,N)=\dim R-\grade(\Ann_RN,M), provided that either projective dimension of M or injective dimension of N is finite. Finally, over such rings, we establish an analogue of the Hartshorne-Lichtenbaum Vanishing Theorem in the context of generalized local cohomology modules.

math.AC↗