The ValabregaValla modules of monomial ideals
Abstract
In this paper, we focus on the initial degree and the vanishing of the ValabregaValla module of a pair of monomials ideals $J\subseteq I$ in a polynomials ring over a field $\mathbb{K}$. We prove that the initial degree of this module is bounded above by the maximum degree of a minimal generators of $J$. For edge ideals of graphs, a complete characterization of the vanishing of the ValabregaValla module is given. For higher degree ideals, we find classes which the ValabregaValla module vanishes. For the case that $J$ is the facet ideal of a clutter $\mathcal{C}$ and $I$ is the defining ideal of singular subscheme of $J$, the nonvanishing of this module is investigated in terms of the combinatorics of $\mathcal{C}$. Finally, we describe the defining ideal of the Rees algebra of $I/J$ provided that the ValabregaValla module is zero.
 Publication:

arXiv eprints
 Pub Date:
 April 2020
 arXiv:
 arXiv:2004.10422
 Bibcode:
 2020arXiv200410422N
 Keywords:

 Mathematics  Commutative Algebra;
 13A30;
 13F55;
 05E40
 EPrint:
 18 Pages, Comments are welcome