Koszulness of binomial edge ideals

Abstract

Let $G$ be a simple graph on the vertex set $V(G)=[n]=\{1,...,n\}$ and edge ideal $E(G)$. We consider the class of closed graphs. A closed graph is a simple graph satisfying the following property: for all edges $\{i,j\}$ and $\{k,\ell \}$ with $i<j$ and $k<\ell$ one has $\{j,\ell \}\in E(G)$ if $i=k$, and $\{i,k\}\in E(G)$ if $j=\ell$. We state some criteria for the closedness of a graph $G$ that do not depend necessarily from the labelling of its vertex set. Consequently, if $S=K[x_1,...,x_n,y_1,...,y_n]$ is a polynomial ring in $2n$ variables with coefficients in a field $K$, we obtain some criteria for the Koszulness of the quotient algebra $S/J_G$, where $J_G$ is the binomial edge ideal of $S$ i.e. the ideal generated by the binomials $f_{ij}=x_i{y}_j-x_j{y}_i$ such that $i<j$ and $\{i,j\}$ is an edge of $G$ (\cite{HH}).

10 pages, 1 figure The title has changed in: Binomial edge ideals with quadratic Gröbner bases

References in corpus (1)

• Distributive Lattices, Bipartite Graphs and Alexander Duality

Cited by in corpus (1)

• Weakly closed graphs and F-purity of binomial edge ideals