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Siamak Yassemi

Publications and source records attributed to Siamak Yassemi.

At least 19 recordsLinked to original sources
arXiv2026

The cones of g-vectors

This paper studies the wall-chamber structures of finite-dimensional ($τ$-tilting infinite) algebras via generic decompositions of g-vectors. In particular, we examine regions outside the chambers. We show that the cones of g-vectors are rational and simplicial. Moreover, we prove that the open cone of a given g-vector coincides with the interior of its $\TF$-equivalence class if and only if the two have the same dimension. Furthermore, we establish that g-vectors satisfy the ray condition when they are sufficiently far from the origin. As an application, we generalize several results of Asai and Iyama concerning $\TF$-equivalence classes of g-vectors.

math.RT↗
arXiv2025

The non-decreasing condition on g-vectors

The non-decreasing condition on g-vectors is introduced. Our study shows that this condition is both necessary and sufficient to ensure that the generically indecomposable direct summands of a given g-vector are linearly independent. Additionally, we prove that for any finite dimensional algebra $Λ$, under the non-decreasing condition, the number of generically indecomposable irreducible components that appear in the decomposition of a given generically $τ$-reduced component is lower than or equal to $|Λ|$. This solves the conjecture concerning the cardinality of component clusters by Cerulli-Labardini-Schröer, in a reasonable generality. Lastly, we study numerical criteria to check the wildness of g-vectors.

math.RT↗
arXiv2024

Lyubeznik tables of $S_r$ and $CM_r$ rings

We describe the shape of the Lyubeznik table of either rings in positive characteristic or Stanley-Reisner rings in any characteristic when they satisfy Serre's condition $S_r$ or they are Cohen-Macaulay in a given codimension, condition denoted by $CM_r$. Moreover we show that these results are sharp.

math.AC↗
arXiv2023

On the index of depth stability of symbolic powers of cover ideals of graphs

Let $G$ be a graph with $n$ vertices and let $S=\mathbb{K}[x_1,\dots,x_n]$ be the polynomial ring in $n$ variables over a field $\mathbb{K}$. Assume that $I(G)$ and $J(G)$ denote the edge ideal and the cover ideal of $G$, respectively. We provide a combinatorial upper bound for the index of depth stability of symbolic powers of $J(G)$. As a consequence, we compute the depth of symbolic powers of cover ideals of fully clique-whiskered graphs. Meanwhile, we determine a class of graphs $G$ with the property that the Castelnuovo--Mumford regularity of $S/I(G)$ is equal to the induced matching number of $G$.

math.AC↗
arXiv2020

Cohen-Macaulay edge-weighted edge ideals of very well-covered graphs

We characterize unmixed and Cohen-Macaulay edge-weighted edge ideals of very well-covered graphs. We also provide examples of oriented graphs which have unmixed and non-Cohen-Macaulay vertex-weighted edge ideals, while the edge ideal of their underlying graph is Cohen-Macaulay. This disproves a conjecture posed by Pitones, Reyes and Toledo.

math.AC↗
arXiv2019

Improved bounds for the regularity of powers of edge ideals of graphs

Let $G$ be a graph with edge ideal $I(G)$. We recall the notions of $\min-match_{\{K_2, C_5\}}(G)$ and $\ind-match_{\{K_2, C_5\}}(G)$ from \cite{sy}. We show that $${\rm reg}(I(G)^s)\leq 2s+\min-match_{\{K_2, C_5\}}(G)-1,$$for all $s\geq 1$, which implies that$${\rm reg}(I(G)^s)\leq 2s+\min-match(G)-1.$$Moreover, we show that$${\rm reg}(I(G)^s)\geq 2s+\ind-match_{\{K_2, C_5\}}(G)-2,$$and if $\ind-match_{\{K_2, C_5\}}(G)$ is an odd integer, then$${\rm reg}(I(G)^s)\geq 2s+\ind-match_{\{K_2, C_5\}}(G)-1.$$Furthermore, it is shown that$${\rm reg}(I(G)^s)\leq 2s+\ord-match(G)-1,$$where $\ord-match(G)$ denotes the ordered matching number of $G$. Finally, we construct infinitely many connected graphs which satisfy the following strict inequalities:$$2s+\ind-match(G)-1 < {\rm reg}(I(G)^s)< 2s+{\rm cochord}(G)-1.$$This gives a positive answer to a question asked in \cite{jns}.

math.AC↗
arXiv2019

Infinitely Generated Gorenstein Tilting Modules

The theory of finitely generated relative (co)tilting modules has been established in the 1980s by Auslander and Solberg, and infinitely generated relative tilting modules have recently been studied by many authors in the context of Gorenstein homological algebra. In this work, we build on the theory of infinitely generated Gorenstein tilting modules by developing "Gorenstein tilting approximations" and employing these approximations to study Gorenstein tilting classes and their associated relative cotorsion pairs. As applications of our results, we discuss the problem of existence of complements to partial Gorenstein tilting modules as well as some connections between Gorenstein tilting modules and finitistic dimension conjectures.

math.RT↗
arXiv2019

Cohen-Macaulay homological dimensions

We introduce new homological dimensions, namely the Cohen-Macaulay projective, injective and flat dimensions for homologically bounded complexes. Among other things we show that (a) these invariants characterize the Cohen-Macaulay property for local rings, (b) Cohen-Macaulay flat dimension fits between the Gorenstein flat dimension and the large restricted flat dimension, and (c) Cohen-Macaulay injective dimension fits between the Gorenstein injective dimension and the Chouinard invariant.

math.AC↗
arXiv2018

Graded Betti numbers of good filtrations

The asymptotic behavior of graded Betti numbers of powers of homogeneous ideals in a polynomial ring over a field has recently been reviewed. We extend quasi polynomial behavior of graded Betti numbers of powers of homogenous ideals to Z-graded algebra over Notherian local ring. Furthermore our main result treats the Betti table of filtrations which is finite or integral over the Rees algebra.

math.AC↗
arXiv2018

Tilting Modules Under Special Base Changes

Given a non-unit, non-zero-divisor, central element $x$ of a ring $Λ$, it is well known that many properties or invariants of $Λ$ determine, and are determined by, those of $Λ/ x Λ$ and $Λ_x$. In the present paper, we investigate how the property of "being tilting" behaves in this situation. It turns out that any tilting module over $Λ$ gives rise to tilting modules over $Λ_x$ and $Λ/ x Λ$ after localization and passing to quotient respectively. On the other hand, it is proved that under some mild conditions, a module over $Λ$ is tilting if its corresponding localization and quotient are tilting over $Λ_x$ and $Λ/ x Λ$ respectively.

math.RT↗
arXiv2017

Regularity of Powers of edge ideal of very well-covered graphs

Let $k\geq 3$ be an integer and $G$ be a very well-covered graph with ${\rm odd-girth}(G)\geq 2k+1$. Assume that $I(G)$ is the edge ideal of $G$. We show that for every integer $s$ with $1\leq s\leq k-2$, we have ${\rm reg}(I(G)^s)=2s+ν(G)-1$, where $ν(G)$ is the induced matching number of $G$.

math.AC↗
arXiv2017

On the regularity of edge ideal of graphs

Let $G$ be a graph with $n$ vertices, $S=\mathbb{K}[x_1,\dots,x_n]$ be the polynomial ring in $n$ variables over a field $\mathbb{K}$ and $I(G)$ denote the edge ideal of $G$. For every collection $\mathcal{H}$ of connected graphs with $K_2\in \mathcal{H}$, we introduce the notions of $\ind-match_{\mathcal{H}}(G)$ and $\min-match_{\mathcal{H}}(G)$. It will be proved that the inequalities $\ind-match_{\{K_2, C_5\}}(G)\leq{\rm reg}(S/I(G))\leq\min-match_{\{K_2, C_5\}}(G)$ are true. Moreover, we show that if $G$ is a Cohen--Macaulay graph with girth at least five, then ${\rm reg}(S/I(G))=\ind-match_{\{K_2, C_5\}}(G)$. Furthermore, we prove that if $G$ is a paw--free and doubly Cohen--Macaulay graph, then ${\rm reg}(S/I(G))=\ind-match_{\{K_2, C_5\}}(G)$ if and only if every connected component of $G$ is either a complete graph or a $5$-cycle graph. Among other results, we show that for every doubly Cohen--Macaulay simplicial complex, the equality ${\rm reg}(\mathbb{K}[Δ])={\rm dim}(\mathbb{K}[Δ])$ holds.

math.AC↗
arXiv2016

Cohen-Macaulay lexsegment complexes in arbitrary codimension

We characterize pure lexsegment complexes which are Cohen-Macaulay in arbitrary codimension. More precisely, we prove that any lexsegment complex is Cohen-Macaulay if and only if it is pure and its one dimensional links are connected, and, a lexsegment flag complex is Cohen-Macaulay if and only if it is pure and connected. We show that any non-Cohen-Macaulay lexsegment complex is a Buchsbaum complex if and only if it is a pure disconnected flag complex. For $t\ge 2$, a lexsegment complex is strictly Cohen-Macaulay in codimension $t$ if and only if it is the join of a lexsegment pure disconnected flag complex with a $(t-2)$-dimensional simplex. When the Stanley-Reisner ideal of a pure lexsegment complex is not quadratic, the complex is Cohen-Macaulay if and only if it is Cohen-Macaulay in some codimension. Our results are based on a characterization of Cohen-Macaulay and Buchsbaum lexsegment complexes by Bonanzinga, Sorrenti and Terai.

math.AC↗
arXiv2015

The behaviors of expansion functor on monomial ideals and toric rings

In this paper we study some algebraic and combinatorial behaviors of expansion functor. We show that on monomial ideals some properties like polymatroidalness, weakly polymatroidalness and having linear quotients are preserved under taking the expansion functor. The main part of the paper is devoted to study of toric ideals associated to the expansion of subsets of monomials which are minimal with respect to divisibility. It is shown that, for a given discrete polymatroid $P$, if toric ideal of $P$ is generated by double swaps then toric ideal of any expansion of $P$ has such a property. This result, in a special case, says that White's conjecture is preserved under taking the expansion functor. Finally, the construction of Gröbner bases and some homological properties of toric ideals associated to expansions of subsets of monomials is investigated.

math.AC↗
arXiv2014

Generalized mixed product ideals

We consider classes of ideals which generalize the mixed product ideals introduced by Restuccia and Villarreal, and also generalize the expansion construction by Bayati and the first author \cite{BH}. We compute the minimal graded free resolution of generalized mixed product ideals and show that the regularity of a generalized mixed product ideal coincides with regularity of the monomial ideal by which it is induced.

math.AC↗