Morse resolutions of monomial ideals and Betti splittings

TL;DR

The paper develops a Morse-theoretic framework for free resolutions of monomial ideals by introducing pruned resolutions, which arise from acyclic matchings on the Taylor (and related) complexes. It unifies classical constructions such as the Eliahou-Kervaire and Herzog-Takayama resolutions within this framework and integrates Betti splittings to enable recursive computation of Betti numbers. The authors prove minimality results for several broad classes (including vertex-splittable, linear-quotients, $p$-Borel, and $\mathbf{a}$-stable ideals) and extend the approach to powers and to graph- and hypergraph-related ideals, using recursive reductions via subideals and Artinian-type splittings. Overall, the work provides a cohesive Morse-theoretic toolkit for constructing minimal free resolutions across diverse monomial-ideal families, with implications for both algebraic and combinatorial topology aspects of the subject.

Abstract

We use discrete Morse theory to study free resolutions of monomial ideals in combination with splitting techniques. We establish the minimality of such pruned resolutions for several classes of ideals, including stable and linear quotient ideals. In particular, we unify classical constructions such as the Eliahou-Kervaire and Herzog-Takayama resolutions within the pruned resolution framework. Additionally, we introduce methods to reduce the minimality study of a pruned resolution for an ideal to that of a smaller subideal and present a variant of our pruned resolution for powers of monomial ideals.

Theorems & Definitions (71)

• Theorem 1
• Theorem 2
• Corollary 1
• Theorem 3
• Theorem 4
• Proposition 2.1: BatziesWelker
• Theorem 2.2: BatziesWelker
• Theorem 3.2
• Remark 3.3
• Remark 3.4