Searcharxiv⌕ Search

arXiv subjects

Siamak Yassemi

Publications and source records attributed to Siamak Yassemi.

At least 37 records · Page 2Linked to original sources
arXiv2013

A new construction for Cohen-Macaulay graphs

Let $G$ be a finite simple graph on a vertex set $V(G)=\{x_{11}, \ldots, x_{n1}\}$. Also let $m_1, \ldots,m_n \geq 2$ be integers and $G_1, \ldots, G_n$ be connected simple graphs on the vertex sets $V(G_i)=\{x_{i1}, \ldots, x_{im_i}\}$. In this paper, we provide necessary and sufficient conditions on $G_1, \ldots, G_n$ for which the graph obtained by attaching $G_i$ to $G$ is unmixed or vertex decomposable. Then we characterize Cohen--Macaulay and sequentially Cohen--Macaulay graphs obtained by attaching the cycle graphs or connected chordal graphs to an arbitrary graphs.

math.AC↗
arXiv2013

Free resolution of powers of monomial ideals and Golod rings

Let $S = \mathbb{K}[x_1, \dots, x_n]$ be the polynomial ring over a field $\mathbb{K}$. In this paper we present a criterion for componentwise linearity of powers of monomial ideals. In particular, we prove that if a square-free monomial ideal $I$ contains no variable and some power of $I$ is componentwise linear, then $I$ satisfies gcd condition. For a square-free monomial ideal $I$ which contains no variable, we show that $S/I$ is a Golod ring provided that for some integer $s\geq 1$, the ideal $I^s$ has linear quotient with respect to a monomial order. We also provide a lower bound for some Betti numbers of powers of a square-free monomial ideal which is generated in a single degree.

math.AC↗
arXiv2013

Cohen-Macaulay-ness in codimension for bipartite graphs

Let $G$ be an unmixed bipartite graph of dimension $d-1$. Assume that $K_{n,n}$, with $n\ge 2$, is a maximal complete bipartite subgraph of $G$ of minimum dimension. Then $G$ is Cohen-Macaulay in codimension $d-n+1$. This generalizes a characterization of Cohen-Macaulay bipartite graphs by Herzog and Hibi and a result of Cook and Nagel on unmixed Buchsbaum graphs. Furthermore, we show that any unmixed bipartite graph $G$ which is Cohen-Macaulay in codimension $t$, is obtained from a Cohen-Macaulay graph by replacing certain edges of $G$ with complete bipartite graphs. We provide some examples.

math.AC↗
arXiv2012

Relative Tor functors with respect to a semidualizing module

We consider relative Tor functors built from resolutions described by a semidualizing module C over a commutative noetherian ring R. We show that the bifunctors Tor^{F_CM}_i (-,-) and Tor^{P_CM}_i (-,-), defined using flat-like and projective-like resolutions, are isomorphic. We show how the vanishing of these functors characterizes the finiteness of the homological dimension F_C-pd, and we use this to give a relation between the F_C-pd of a given module and that of a pure submodule. On the other hand, we show that other relations that one may expect to hold similarly, fail in general. In fact, such relations force the semidualizing modules under consideration to be trivial.

math.AC↗
arXiv2011

Homology of powers of ideals: Artin--Rees numbers of syzygies and the Golod property

For an ideal I in a regular local ring (R,m)$ with residue class field K = R/m or a standard graded K-algebra R we show that for k >> 0 --> the Artin--Rees number of the syzygy modules of I^k as submodules of the free modules from a free resolution is constant, and thereby present the Artin-Rees number as a proper replacement of regularity in the local situation, --> the ring R/I^k is Golod, its Poincer{é}-Betti series is rational and the Betti numbers of the free resolution of K over R/I^k are polynomials in k of a specific degree. The first result is an extension of work of Kodiyalam and Cutkosky, Herzog & Trung on the regularity of I^k for k >> 0 from the graded situation to the local situation. The polynomiality consequence of the second result is an analog of work by Kodiyalam on the behavior of Betti numbers of the minimal free resolution of R/I^k over R.

math.AC↗
arXiv2010

$M$-Shellability of Discrete Polymatroids

In this note we show that every discrete polymatroid is $M$-shellable. This gives, in a partial case, a positive answer to a conjecture of Chari and improves a recent result of Schweig where he proved that the $h$-vector of a lattice path matroid satisfies a conjecture of Stanley.

math.CO↗
arXiv2010

Depth formula via complete intersection flat dimension

We prove the depth formula, for homologically bounded complexes $X, Y$ provided that the complete intersection flat dimension of $X$ is finite and $\sup(X\utp_RY)<\infty$. In particular, let $M$ and $N$ are two $R$-modules and the complete intersection flat dimension of $M$ is finite. Then $M$ and $N$ satisfies the depth formula, provided $\Tor^R_i(M,N)=0$ for all $i\ge 1$.

math.AC↗
arXiv2010

Vertex decomposability and regularity of very well-covered graphs

A graph $G$ is well-covered if it has no isolated vertices and all the maximal independent sets have the same cardinality. If furthermore two times this cardinality is equal to $|V(G)|$, the graph $G$ is called very well-covered. The class of very well-covered graphs contains bipartite well-covered graphs. Recently in \cite{CRT} it is shown that a very well-covered graph $G$ is Cohen-Macaulay if and only if it is pure shellable. In this article we improve this result by showing that $G$ is Cohen-Macaulay if and only if it is pure vertex decomposable. In addition, if $I(G)$ denotes the edge ideal of $G$, we show that the Castelnuovo-Mumford regularity of $R/I(G)$ is equal to the maximum number of pairwise 3-disjoint edges of $G$. This improves Kummini's result on unmixed bipartite graphs.

math.AC↗
arXiv2010

A note on the regularity of products

Let $S={\Bbb K}[x_1,\dots,x_n]$ denote a polynomial ring over a field $\Bbb K$. Given a monomial ideal $I$ and a finitely generated multigraded $M$ over $S$, we follow Herzog's method to construct a multigraded free $S$-resolution of $M/IM$ by using multigraded $S$-free resolutions of $S/I$ and $M$. The complex constructed in this paper is used to prove the inequality $\Reg(IM)\leq \Reg(I)+\Reg(M)$ for a large class of ideals and modules. In the case where $M$ is an ideal, under one relative condition on the generators which specially does not involve the dimensions, the inequality $\Reg(IM)\leq \Reg(I)+\Reg(M)$ is proven.

math.AC↗
arXiv2010

Sequentially $S_r$ simplicial complexes and sequentially $S_2$ graphs

We introduce sequentially $S_r$ modules over a commutative graded ring and sequentially $S_r$ simplicial complexes. This generalizes two properties for modules and simplicial complexes: being sequentially Cohen-Macaulay, and satisfying Serre's condition $S_r$. In analogy with the sequentially Cohen-Macaulay property, we show that a simplicial complex is sequentially $S_r$ if and only if its pure $i$-skeleton is $S_r$ for all $i$. For $r=2$, we provide a more relaxed characterization. As an algebraic criterion, we prove that a simplicial complex is sequentially $S_r$ if and only if the minimal free resolution of the ideal of its Alexander dual is componentwise linear in the first $r$ steps. We apply these results for a graph, i.e., for the simplicial complex of the independent sets of vertices of a graph. We characterize sequentially $S_r$ cycles showing that the only sequentially $S_2$ cycles are odd cycles and, for $r\ge 3$, no cycle is sequentially $S_r$ with the exception of cycles of length 3 and 5. We extend certain known results on sequentially Cohen-Macaulay graphs to the case of sequentially $S_r$ graphs. We prove that a bipartite graph is vertex decomposable if and only if it is sequentially $S_2$. We provide some more results on certain graphs which in particular implies that any graph with no chordless even cycle is sequentially $S_2$. Finally, we propose some questions.

math.AC↗
arXiv2010

On the existence of embeddings into modules of finite homological dimensions

Let R be a commutative Noetherian local ring. We show that R is Gorenstein if and only if every finitely generated R-module can be embedded in a finitely generated R-module of finite projective dimension. This extends a result of Auslander and Bridger to rings of higher Krull dimension, and it also improves a result due to Foxby where the ring is assumed to be Cohen-Macaulay.

math.AC↗
arXiv2010

Rings whose total graphs have genus at most one

Let $R$ be a commutative ring with $\Z(R)$ its set of zero-divisors. In this paper, we study the total graph of $R$, denoted by $\T(Γ(R))$. It is the (undirected) graph with all elements of $R$ as vertices, and for distinct $x, y\in R$, the vertices $x$ and $y$ are adjacent if and only if $x + y\in\Z(R)$. We investigate properties of the total graph of $R$ and determine all isomorphism classes of finite commutative rings whose total graph has genus at most one (i.e., a planar or toroidal graph). In addition, it is shown that, given a positive integer $g$, there are only finitely many finite rings whose total graph has genus $g$.

math.AC↗
arXiv2010

Bipartite $S_2$ graphs are Cohen-Macaulay

In this paper we show that if the Stanley-Reisner ring of the simplicial complex of independent sets of a bipartite graph $G$ satisfies Serre's condition $S_2$, then $G$ is Cohen-Macaulay. As a consequence, the characterization of Cohen-Macaulay bipartite graphs due to Herzog and Hibi carries over this family of bipartite graphs. We check that the equivalence of Cohen-Macaulay property and the condition $S_2$ is also true for chordal graphs and we classify cyclic graphs with respect to the condition $S_2$.

math.AC↗
arXiv2010

Rings that are homologically of minimal multiplicity

Let R be a local Cohen-Macaulay ring with canonical module ω_R. We investigate the following question of Huneke: If the sequence of Betti numbers \{β_i^R(ω_R)\} has polynomial growth, must R be Gorenstein? This question is well-understood when R has minimal multiplicity. We investigate this question for a more general class of rings which we say are homologically of minimal multiplicity. We provide several characterizations of the rings in this class and establish a general ascent and descent result.

math.AC↗
arXiv2009

A generalization of $k$-Cohen-Macaulay complexes

For a positive integer $k$ and a non-negative integer $t$ a class of simplicial complexes, to be denoted by $k$-${\rm CM}_t$, is introduced. This class generalizes two notions for simplicial complexes: being $k$-Cohen-Macaulay and $k$-Buchsbaum. In analogy with the Cohen-Macaulay and Buchsbaum complexes, we give some characterizations of ${\rm CM}_t(=$1-${\rm CM}_t)$ complexes, in terms of vanishing of some homologies of its links and, in terms of vanishing of some relative singular homologies of the geometric realization of the complex and its punctured space. We show that a complex is $k$-${\rm CM}_t$ if and only if the links of its nonempty faces are $k$-${\rm CM}_{t-1}$. We prove that for an integer $s\le d$, the $(d-s-1)$-skeleton of a $(d-1)$-dimensional $k$-${\rm CM}_t$ complex is $(k+s)$-${\rm CM}_t$. This result generalizes Hibi's result for Cohen-Macaulay complexes and Miyazaki's result for Buchsbaum complexes.

math.AC↗