Koszulness, Krull dimension, and other properties of graph-related algebras

  • Authors:

  • Alexandru Constantinescu;Matteo Varbaro

  • Affiliations:

  • Dipartimento di Matematica, Università degli Studi di Genova, Genova, Italy 16146;Dipartimento di Matematica, Università degli Studi di Genova, Genova, Italy 16146

  • Venue:
  • Journal of Algebraic Combinatorics: An International Journal
  • Year:
  • 2011

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Abstract

The algebra of basic covers of a graph G, denoted by $\bar{A}(G)$ , was introduced by Herzog as a suitable quotient of the vertex cover algebra. In this paper we compute the Krull dimension of $\bar{A}(G)$ 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 $\bar{A}(G)$ 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.