The ${\rm N}_{2,p}$-property of binomial extensions of simplicial complexes

Abstract

M. Morales introduced a family of binomial ideals that are binomial extensions of square free monomial ideals. Let $I\subset \si$ be a square free monomial ideal and $J\subset\sis$ a sum of scroll ideals with some extra conditions, we define the binomial extension of $I$ as $\B=I+J\subset \sis$. We set $p_2(\B)$ the minimal $i\in\N$ such that there exists $j>2$ such that $\beta_{i,i+j}(\B)\neq 0$. In the case where J=0, Fr\"oberg characterized combinatorally the case $p_2(I)=\infty$; later Eisenbud et al. solved the case $p_2(I)<\infty$. We obtain a similar result as Fr\"oberg for the binomial extensions and we find lower and upper bounds of $p_2(\B)$ for some families of binomial extensions in combinatorial terms as Eisenbud et al. With some additional hypothesis we can compute $p_2(\B)$.