The algebra of basic covers of a graph G, denoted by A(G), was introduced by Herzog as a suitable quotient of the vertex cover algebra. In this paper 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. Furthermore, we show that if the graph is bipartite, then A(G) is a homogeneous algebra with straightening laws, and thus it is Koszul. Finally, we characterize the Cohen-Macaulay property and the Castelnuovo-Mumford regularity of the edge ideal of a certain class of graphs. © 2011 Springer Science+Business Media, LLC.
Koszulness, Krull dimension, and other properties of graph-related algebras
Constantinescu A.;Varbaro M.
2011-01-01
Abstract
The algebra of basic covers of a graph G, denoted by A(G), was introduced by Herzog as a suitable quotient of the vertex cover algebra. In this paper 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. Furthermore, we show that if the graph is bipartite, then A(G) is a homogeneous algebra with straightening laws, and thus it is Koszul. Finally, we characterize the Cohen-Macaulay property and the Castelnuovo-Mumford regularity of the edge ideal of a certain class of graphs. © 2011 Springer Science+Business Media, LLC.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
