Related papers: When is a Squarefree Monomial Ideal of Linear Type…

Let $S=K[x_1,\ldots,x_n]$ be the polynomial ring over a field and $A$ a standard graded $S$-algebra. In terms of the Gr\"obner basis of the defining ideal $J$ of $A$ we give a condition, called the x-condition, which implies that all graded…

In this paper, we study toric ideals generated by circuits. For toric ideals which have squarefree quadratic initial ideals, a sufficient condition to be generated by circuits is given. In particular, squarefree Veronese subrings, the…

Let $R=K[x_1,\ldots,x_n]$ be the polynomial ring in $n$ variables over a field $K$. We show that if $G$ is a connected graph with a basic $5$-cycle $C$, then $G$ is a sequentially Cohen-Macaulay graph if and only if there exists a shedding…

We consider the edge ideals of large classes of graphs with whiskers and for these ideals we prove that the arithmetical rank is equal to the big height. Then we extend these results to other classes of squarefree monomial ideals, generated…

We adapt the notion of an algebraic theory to work in the setting of quasicategories developed recently by Joyal and Lurie. We develop the general theory at some length. We study one extended example in detail: the theory of commutative…

The goal of this paper is to present examples of families of homogeneous ideals in the polynomial ring over a field that satisfy the following condition: every product of ideals of the family has a linear free resolution. As we will see,…

Nagel and R\"omer introduced the class of weakly vertex decomposable simplicial complexes, which include matroid, shifted, and Gorenstein complexes as well as vertex decomposable complexes. They proved that the Stanley-Reisner ideal of…

We prove a sufficient and a necessary condition for a square-free monomial ideal $J$ associated to a (dual) hypergraph to have projective dimension equal to the minimal number of generators of $J$ minus 2. We also provide an effective…

Each monomial ideal over a polynomial ring admits a free resolution which has the structure of a DG-algebra, namely, the Taylor resolution. A pivot resolution of a monomial ideal, which we introduce, is a resolution that is always shorter…

In this article we investigate when a homogeneous ideal in a graded ring is normal, that is, when all positive powers of the ideal are integrally closed. We are particularly interested in homogeneous ideals in an N-graded ring generated by…

We study monomial ideals using the operation polarization to first turn them into square-free monomial ideals. We focus on monomial ideals whose polarization produce simplicial trees, and show that many of the properties of simplicial trees…

An equigenerated monomial ideal $I$ in the polynomial ring $R=k[z_1,\ldots, z_n]$ is a Freiman ideal if $\mu(I^2)=\ell(I)\mu(I)-{\ell(I)\choose 2}$ where $\ell(I)$ is the analytic spread of $I$ and $\mu(I)$ is the number of minimal…

For a monomial ideal $I$ of a polynomial ring $S$, a "polarization" of $I$ is a \textit{squarefree} monomial ideal $J$ of a larger polynomial ring $S'$ such that $S/I$ is a quotient of $S'/J$ by a regular sequence (consisting of degree 1…

Let $I$ be a monomial ideal in a polynomial ring $S=K[x_1,\ldots,x_n]$ over a field $K$ with $n=2$ or $3$, and let $\overline{I}$ be its integral closure. We will show that $\text{reg} (\overline{I}) \le \text{reg} (I)$. Furthermore, if $I$…

We prove that hypersurfaces defined by irreducible square-free polynomials have rational singularities. As an easy consequence, we deduce that certain (possibly non-square-free) polynomials associated to pairs of square-free polynomials…

The purpose of this paper is to give an elementary proof to the theorem due to Avramov on certain determinantal ideals of linear type.

In this paper we study the normality of monomial ideals using linear programming and graph theory. We give normality criteria for monomial ideals, for ideals generated by monomials of degree two, and for edge ideals of graphs and clutters…

In this paper, we study Cstelnuovo-Mumford regularity of square-free monomial ideals generated in degree 3. We define some operations on the clutters associated to such ideals and prove that the regularity is conserved under these…

Let $I,J$ be componentwise linear ideals in a polynomial ring $S$. We study necessary and sufficient conditions for $I+J$ to be componentwise linear. We provide a complete characterization when $\dim S=2$. As a consequence, any…

We develop a new technique for studying monomial ideals in the standard polynomial rings $A[X_1,\ldots,X_d]$ where $A$ is a commutative ring with identity. The main idea is to consider induced ideals in the semigroup ring…