Dimension, depth and zero-divisors of the algebra of basic $k$-covers of a graph

Abstract

We study the basic $k$-covers of a bipartite graph $G$; the algebra $\AG$ they span, first studied by Herzog, is the fiber cone of the Alexander dual of the edge ideal. We characterize when $\AG$ is a domain in terms of the combinatorics of $G$; if follows from a result of Hochster that when $\AG$ is a domain, it is also Cohen-Macaulay. We then study the dimension of $\AG$ by introducing a geometric invariant of bipartite graphs, the "graphical dimension". We show that the graphical dimension of $G$ is not larger than $\dim(\AG)$, and equality holds in many cases (e.g. when $G$ is a tree, or a cycle). Finally, we discuss applications of this theory to the arithmetical rank.

31 pages

Cited by in corpus (3)

• On the Hilbert series of vertex cover algebras of Cohen-Macaulay bipartite graphs
• Unmixed Graphs that are Domains
• On the Hilbert series of vertex cover algebras of unmixed bipartite graphs