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Siamak Yassemi

Publications and source records attributed to Siamak Yassemi.

At least 55 records · Page 3Linked to original sources
arXiv2009

The complement of a connected bipartite graph is vertex decomposable

Associated to a simple undirected graph $G$ is a simplicial complex $Δ_G$ whose faces correspond to the independent sets of $G$. A graph $G$ is called vertex decomposable if $Δ_G$ is a vertex decomposable simplicial complex. We are interested in determining what families of graph have the property that the complement of $G$, denoted by $\overline{G}$, is vertex decomposable. We obtain the result that the complement of a connected bipartite graph is vertex decomposable and so it is Cohen-Macaulay due to pureness of $Δ_{\overline{G}}$.

math.AC↗
arXiv2008

Homological flat dimensions

For finitely generated module $M$ over a local ring $R$, the conventional notions of complete intersection dimension $\cid_R M$ and Cohen-Macaulay dimension $\cmdim_R M$ do not extend to cover the case of infinitely generated modules. In this paper we introduce similar invariants for not necessarily finitely generated modules, (namely, complete intersection flat and Cohen-Macaulay flat dimensions) which for finitely generated modules, coincide with the corresponding classical ones.

math.AC↗
arXiv2007

Comaximal graph of commutative rings

Let $R$ be a commutative ring with identity. Let $Γ(R)$ be a graph with vertices as elements of $R$, where two distinct vertices $a$ and $b$ are adjacent if and only if $Ra+Rb=R$. In this paper we consider a subgraph $Γ_2(R)$ of $Γ(R)$ which consists of non-unit elements. We look at the connectedness and the diameter of this graph. We completely characterize the diameter of the graph $Γ_2(R)\setminus\J(R)$. In addition, it is shown that for two finite semi-local rings $R$ and $S$, if $R$ is reduced, then $Γ(R)\congΓ(S)$ if and only if $R\cong S$.

math.AC↗
arXiv2007

Complete $r$-partite zero-divisor graphs and coloring of commutative semigroups

For a commutative semigroup $S$ with 0, the zero-divisor graph of $S$ denoted by $Γ(S)$ is the graph whose vertices are nonzero zero-divisor of $S$, and two vertices $x$, $y$ are adjacent in case $xy=0$ in $S$. In this paper we study the case where the graph $Γ(S)$ is complete $r$-partite for a positive integer $r$. Also we study the commutative semigroups which are finitely colorable.

math.GR↗
arXiv2007

Stanley decompositions and partitionable simplicial complexes

We study Stanley decompositions and show that Stanley's conjecture on Stanley decompositions implies his conjecture on partitionable Cohen-Macaulay simplicial complexes. We also prove these conjectures for all Cohen-Macaulay monomial ideals of codimension 2 and all Gorenstein monomial ideals of codimension 3.

math.AC↗
arXiv2006

Local cohomology and Gorenstein injective dimension over local homomorphisms

A generalization of Grothendieck's non-vanishing theorem is proved for a module which is finite over a local homomorphism. It is also proved that the Gorenstein injective dimension of such a module, if finite, is bounded below by its Krull dimension and is equal to the supremum of the depths of the localizations of the ring over primes in the support of the module.

math.AC↗
arXiv2005

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 $\Ext^2_R(R/\fa,\H^t_\fa(M))$.

math.AC↗
arXiv2005

Characterizing local rings via homological dimensions and regular sequences

Let (R,m) be a Noetherian local ring of depth d and C a semidualizing R-complex. Let M be a finite R-module and t an integer between 0 and d. If G_C-dimension of M/IM is finite for all ideals I generated by an R-regular sequence of length at most d-t then either G_C-dimension of M is at most t or C is a dualizing complex. Analogous results for other homological dimensions are also given.

math.AC↗
arXiv2005

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 $i\geq r$.

math.AC↗
arXiv2004

Gorenstein injective dimension, Bass formula and Gorenstein rings

Let $(R,\frak m, k)$ be a noetherian local ring. It is well-known that $R$ is regular if and only if the injective dimension of $k$ is finite. In this paper it is shown that $R$ is Gorenstein if and only if the Gorenstein injective dimension of $k$ is finite. On the other hand a generalized version of the so-called Bass formula is proved for finitely generated modules of finite Gorenstein injective dimension. It also improves Christensen's generalized Bass formula (cf. "Gorenstein dimensions", volume 1747 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2000).

math.AC↗
arXiv2004

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 $M$.

math.AC↗
arXiv2004

Special homological dimensions and Intersection Theorem

Let $(R,\fm)$ be commutative Noetherian local ring. It is shown that $R$ is Cohen--Macaulay ring if there exists a Cohen--Macaulay finite (i.e. finitely generated) $R$--module with finite upper Gorenstein dimension. In addition, we show that, in the Intersection Theorem, projective dimension can be replaced by quasi--projective dimension.

math.AC↗
arXiv2004

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 $\inf{\mathbf R}Γ_{\fa}({\mathbf R}\Hom_R(X,Y))$ 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.

math.AC↗
arXiv2003

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.

math.AC↗
arXiv2003

Tensor Products of Some Special Rings

In this paper we solve a problem, originally raised by Grothendieck, on the properties, i.e. Complete intersection, Gorenstein, Cohen--Macaulay, that are conserved under tensor product of algebras over a field $k$.

math.AC↗
arXiv2002

Gorenstein Dimensions under Base Change

The so-called 'change-of-ring' results are well-known expressions which present several connections between projective, injective and flat dimensions over the various base rings. In this note we extend these results to the Gorenstein dimensions over Cohen-Macaulay local rings.

math.AC↗