The Scarf complex and betti numbers of powers of extremal ideals

TL;DR

This paper investigates the Betti numbers and free resolutions of powers of extremal square-free monomial ideals, providing bounds and explicit descriptions of their minimal free resolutions using the Scarf complex.

Contribution

It introduces the extremal ideal al E_q, which bounds Betti numbers of powers of any ideal generated by q square-free monomials, and explicitly describes their minimal free resolutions for certain cases.

Findings

al E_q provides upper bounds on Betti numbers for powers of square-free monomial ideals.

Minimal free resolutions of al E_q powers are supported on the Scarf complex when q  4 or r  2.

Bounds on projective dimension are established, e.g., pd(I^r) 5 for q 4 and all r
1.

Abstract

This paper is concerned with finding bounds on betti numbers and describing combinatorially and topologically (minimal) free resolutions of powers of ideals generated by a fixed number $q$ of square-free monomials. Among such ideals, we focus on a specific ideal $\mathcal{E}_q$, which we call {\it extremal}, and which has the property that for each $r\ge 1$ the betti numbers of ${\mathcal{E}_q}^r$ are an upper bound for the betti numbers of $I^r$ for any ideal $I$ generated by $q$ square-free monomials (in any number of variables). We study the Scarf complex of the ideals ${\mathcal{E}_q}^r$ and use this simplicial complex to extract information on minimal free resolutions. In particular, we show that ${\mathcal{E}_q}^r$ has a minimal free resolution supported on its Scarf complex when $q\leq 4$ or when $r\leq 2$, and we describe explicitly this complex. For any $q$ and $r$, we also show that $\beta_1({\mathcal{E}_q}^r)$ is the smallest possible, or in other words equal to the number of edges in the Scarf complex. These results lead to effective bounds on the betti numbers of $I^r$, with $I$ as above. For example, we obtain that pd$(I^r)\leq 5$ for all ideals $I$ generated by $4$ square-free monomials and any $r\geq 1$.