Normal Rees algebras arising from vertex decomposable simplicial complexes

TL;DR

This paper proves that Rees algebras of certain ideals derived from vertex decomposable simplicial complexes are normal and Cohen-Macaulay, leading to new normality results for various classes of monomial ideals.

Contribution

It establishes the normality and Cohen-Macaulay property of Rees algebras for ideals associated with vertex decomposable complexes, extending to several classes of monomial ideals.

Findings

Rees algebra of $I_{ riangle^{ ext{vee}}}$ is normal and Cohen-Macaulay for vertex decomposable complexes.

Any squarefree weakly polymatroidal ideal is normal.

Normal ideals are identified among cover and edge ideals of graphs and hypergraphs.

Abstract

We show that for a vertex decomposable simplicial complex $\Delta$, the Rees algebra of $I_{\Delta^{\vee}}$ is a normal Cohen-Macaulay domain. As consequences, we show that any squarefree weakly polymatroidal ideal is normal and we obtain normal ideals among several interesting families of monomial ideals such as cover ideals of graphs and edge ideals of hypergraphs. Moreover, based on a construction on simplicial complexes given by Biermann and Van Tuyl [2], we present families of normal ideals attached to any squarefree monomial ideal.