Realizing resolutions of powers of extremal ideals

TL;DR

This work investigates powers of square-free monomial ideals through the lens of extremal ideals E_q and their Scarf resolutions. By embedding the Scarf complex S_q^r into a polyhedral framework and developing a general Morse-theoretic method, it proves that E_q^3 has a Scarf resolution for all q and provides sharp, scalable bounds on Betti numbers and projective dimension for the third power of any square-free monomial ideal. The results yield exponentially better Betti-number bounds than Taylor-type bounds in the fixed q regime and establish a concrete combinatorial and geometric description of the relevant complexes. The methods promise broader applicability to higher powers and illuminate the deep connections between combinatorial geometry and free resolutions in commutative algebra.

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 $r,q \geq 1$.

Figures (5)

• Figure 1: A comparison between known bounds for Betti numbers of the third power. The $y$-axis represents the value of the Betti number, while the $x$-axis represents homological degree. In the non exponential scale, our bound is not visible as a consequence of the difference in orders of magnitude. Moreover, graphs for higher $q$ look extremely similar because of the asymptotic behaviour of the three bounds.
• Figure 2: Complexes supporting a free resolution of $I=(xy,yz,zu)$
• Figure 3: \ref{['e:q23']}
• Figure 4: $\mathbb{S}^r_4 = \mathbb{U}^r_4$ for $r = 2, 3$. The top of $\mathbb{U}^3_4$ is the same as all of $\mathbb{U}^2_4$ with $\mathbf{e}_4$ added to each vertex, and similarly for the other three corners of $\mathbb{U}^3_4$ with $\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3$, respectively. See \ref{['e:q34']}. The complexes $\mathbb{S}^r_q$ and $\mathbb{U}^r_q$ are no longer equal for $q \geq 5$ and $r \geq 3$; see \ref{['p:weird', 'e:picture-q5-r3']}.
• Figure 5: A comparison between known bounds for betti numbers

Theorems & Definitions (68)

• Theorem 1.1: Lr
• Conjecture 1.2
• Theorem 1.3: \ref{['t:decrease powers']}
• Theorem 1.4: \ref{['t:r=3']}
• Theorem 1.5: \ref{['t:sharpbound3']}
• Remark 1.6: \ref{['t:c-bound']} and \ref{['t:i-bound']}
• Example 2.1
• Definition 2.2
• Example 2.3
• Lemma 2.4