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

TL;DR

This paper studies the algebraic properties of binomial extensions of simplicial complexes, providing bounds and conditions for certain homological invariants, extending previous results from monomial ideals to binomial ideals.

Contribution

It generalizes Fr"oberg's and Eisenbud et al.'s results to binomial extensions, offering bounds and explicit computations for the invariant p_2(B).

Findings

Established bounds for p_2(B) in binomial extensions.

Extended combinatorial characterizations to binomial ideals.

Provided conditions for exact computation of p_2(B).

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 $p_2(\B)$.