Simplicial Resolutions of Powers of Square-free Monomial Ideals

TL;DR

This paper introduces a new, smaller simplicial resolution for powers of square-free monomial ideals, improving upon the Taylor resolution and providing bounds on Betti numbers.

Contribution

It constructs a minimal, size-efficient resolution for powers of square-free monomial ideals based on generator count and relations, and introduces extremal ideals for bounding Betti numbers.

Findings

New simplicial complexes support smaller resolutions

Resolutions are minimal in special cases

Betti numbers are bounded by those of extremal ideals

Abstract

The Taylor resolution is almost never minimal for powers of monomial ideals, even in the square-free case. In this paper we introduce a smaller resolution for each power of any square-free monomial ideal, which depends only on the number of generators of the ideal. More precisely, for every pair of fixed integers $r$ and $q$, we construct a simplicial complex that supports a free resolution of the $r$-th power of any square-free monomial ideal with $q$ generators. The resulting resolution is significantly smaller than the Taylor resolution, and is minimal for special cases. Considering the relations on the generators of a fixed ideal allows us to further shrink these resolutions. We also introduce a class of ideals called "extremal ideals", and show that the Betti numbers of powers of all square-free monomial ideals are bounded by Betti numbers of powers of extremal ideals. Our results lead to upper bounds on Betti numbers of powers of any square-free monomial ideal that greatly improve the binomial bounds offered by the Taylor resolution.