Koszulness, Krull Dimension and Other Properties of Graph-Related Algebras

TL;DR

This paper explores algebraic properties of graph-related algebras, showing bipartite graphs lead to Koszul algebras, computing their Krull dimension, and analyzing Cohen-Macaulay and regularity properties.

Contribution

It establishes that for bipartite graphs, the algebra (G) is Koszul with straightening laws and provides formulas for Krull dimension based on graph combinatorics.

Findings

(G) is Koszul for bipartite graphs.

Krull dimension of (G) is computed from graph structure.

New bounds on arithmetical rank of certain monomial ideals.

Abstract

The algebra of basic covers of a graph G, denoted by \A(G), was introduced by Juergen Herzog as a suitable quotient of the vertex cover algebra. In this paper we show that if the graph is bipartite then \A(G) is a homogeneous algebra with straightening laws and thus is Koszul. Furthermore, we compute the Krull dimension of \A(G) in terms of the combinatorics of G. As a consequence we get new upper bounds on the arithmetical rank of monomial ideals of pure codimension 2. Finally, we characterize the Cohen-Macaulay property and the Castelnuovo-Mumford regularity of the edge ideal of a certain class of graphs.