Dimension, Depth and Zero-Divisors of the Algebra of Basic k-Covers of a Graph

We study the basic k-covers of a bipartite graph G; the algebra A(G) they span, first studied by Herzog, is the fiber cone of the Alexander dual of the edge ideal. We characterize when A(G) is a domain in terms of the combinatorics of G; if follows from a result of Hochster that when A(G) is a domain, it is also Cohen-Macaulay. We then study the dimension of A(G) by introducing a geometric invariant of bipartite graphs, the “graphical dimension”. We show that the graphical dimension of G is not larger than dim(A(G)), 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.