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

  • Bruno Benedetti TU Berlin
  • Alexandru Constatinescu Università di Genova
  • Matteo Varbaro Università di Genova


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; it 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.

Author Biographies

Bruno Benedetti, TU Berlin
Inst. Mathematics, MA 6-2
TU Berlin
Alexandru Constatinescu, Università di Genova
Dipartimento di Matematica

Matteo Varbaro, Università di Genova
Dipartimento di Matematica