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

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

2010-04-28

arxiv.org/abs/1004.4980v1

J. Algebr. Comb. vol. 34 (2011), pp. 375-400

Mathematics

Commutative Algebra

23 pages

Scientific paper

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.

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