Viviana Ene

Cohen-Macaulay binomial edge ideals

We study the depth of classes of binomial edge ideals and classify all closed graphs whose binomial edge ideal is Cohen--Macaulay.

Monomial ideals and toric rings of Hibi type arising from a finite poset

In this paper we study monomial ideals attached to posets, introduce generalized Hibi rings and investigate their algebraic and homological properties. The main tools to study these objects are Grobner basis theory, the concept of sortability due to Sturmfels and the theory of weakly polymatroidal ideals.

The binomial edge ideal of a pair of graphs

We introduce a class of ideals generated by a set of 2-minors of an ( m × n )-matrix of indeterminates indexed by a pair of graphs. This class of ideals is a natural common generalization of binomial edge ideals and ideals generated by adjacent minors. We determine the minimal prime ideals of such ideals and give a lower bound for their degree of nilpotency. In some special cases we compute their Grobner basis and characterize unmixedness and Cohen–Macaulayness.

Koszul Binomial Edge Ideals

It is shown that if the binomial edge ideal of a graph G defines a Koszul algebra, then G must be chordal and claw free. A converse of this statement is proved for a class of chordal and claw-free graphs.

Rank One Maximal Cohen-Macaulay Modules Over Singularities of Type Y 1 3 + Y 2 3 + Y 3 3 + Y 4 3

We describe, by matrix factorizations, the rank one graded maximal Cohen-Macaulay modules over the hypersurface Y 1 3+Y 2 3+ Y 3 3 + Y 4 3.

Betti numbers for p -stable ideals

Introduction Let Ic be an infinite field of characteristic p > 0, R = k[xl , . , x,] the polynomial ring with the standard grading R = @ Rd. d>O Let I c R be a monomial ideal and G ( I ) a minimal set of monomials generating I. Let u be a monomial. Denote by m(u) the maximum integer i such that xilu. If

Steps in the Classification of Cohen–Macaulay Modules Over Singularities of Type xt + y3

9 divides u but x;+l does not, we write xl ( 1 u. An ideal I is stable if it is monomial and for every monomial u E I and i < m(u) it holds

Ideals Generated by Diagonal 2-Minors

Let k be an algebraically closed field of characteristic ≠ 3 and i, j, t some positive integers such that 1 ≤ i < j < t, i + j ≠ t. Then there exist a finite number of nonisomorphic indecomposable maximal Cohen–Macaulay modules N over k[[x, y]] /(xt + y3) such that N / y N is a direct sum of copies of k[[x]] /(xi), k[[x]] /(xj) and we describe them completely.

Grobner Bases in Commutative Algebra

With a simple graph G on [n], we associate a binomial ideal PG generated by diagonal minors of an n × n matrix X = (xij) of variables. We show that for any graph G, PG is a prime complete intersection ideal and determine the divisor class group of K[X]/PG. By using these ideals, one may find a normal domain with free divisor class group of any given rank.