Realizing resolutions of powers of extremal ideals

TL;DR

This paper investigates the structure of powers of extremal ideals, providing new proofs for the case r=3, and offers bounds on Betti numbers and insights into their minimal free resolutions using combinatorial and geometric methods.

Contribution

It proves the conjecture that powers of extremal ideals have resolutions supported on their Scarf complexes for r=3, and describes their combinatorial structure for all r and q.

Findings

Proved the conjecture for r=3 and all q, showing resolutions supported on Scarf complexes.

Provided exponential bounds on Betti numbers for the third power of square-free monomial ideals.

Described faces of the Scarf complex for powers of extremal ideals across all r and q.

Abstract

Extremal ideals are a class of square-free monomial ideals which dominate and determine many algebraic invariants of powers of all square-free monomial ideals. For example, the $r^{th}$ power ${\mathcal{E}_q}^r$ of the extremal ideal on $q$ generators has the maximum Betti numbers among the $r^{th}$ power of any square-free monomial ideal with $q$ generators. In this paper we study the combinatorial and geometric structure of the (minimal) free resolutions of powers of square-free monomial ideals via the resolutions of powers of extremal ideals. Although the end results are algebraic, this problem has a natural interpretation in terms of polytopes and discrete geometry. Our guiding conjecture is that all powers ${\mathcal{E}_q}^r$ of extremal ideals have resolutions supported on their Scarf simplicial complexes, and thus their resolutions are as small as possible. This conjecture is known to hold for $r \leq 2$ or $q \leq 4$. In this paper we prove the conjecture holds for $r=3$ and any $q\geq 1$ by giving a complete description of the Scarf complex of ${\mathcal{E}_q}^3$. This effectively gives us a sharp bound on the betti numbers and projective dimension of the third power of any square-free momomial ideal. For large $i$ and $q$, our bounds on the $i^{th}$ betti numbers are an exponential improvement over previously known bounds. We also describe a large number of faces of the Scarf complex of ${\mathcal{E}_q}^r$ for any $r,q \geq 1$.