The resolutions of generalized co-letterplace ideals and their powers

TL;DR

The paper develops an explicit, multigraded minimal free resolution for generalized co-letterplace ideals \(\mathcal{L}(P,\mathcal{A};\mathfrak{A})\), distinguishing it from prior constructions. It proves the linear-quotient property and linearity for a broad class of these ideals, and provides a detailed description of the homological shift ideals. A central result is that every power \(\mathcal{L}(P,\mathcal{A})^k\) can be realized as a quotient by a regular sequence of variable differences from a larger generalized co-letterplace ideal, enabling an explicit resolution for all powers. Moreover, the associated simplicial complexes \(\Delta(\mathfrak{A})\) are shown to be either simplicial balls or spheres, with a clear criterion for when the complex is a sphere based on the projective dimension. This work broadens the class of ideals with tractable resolutions and links numerical invariants to topological properties of the associated simplicial complexes.

Abstract

We present a natural and explicit multigraded minimal free resolution for each generalized co-letterplace ideal (see Definition 1.1). Our resolution differs significantly from the ones presented in the works of Ene et al. \cite{EHM} and D'Al{`ı} et al. \cite{DFN}. Additionally, we show that each power of a large class of generalized co-letterplace ideals can be represented as the quotient of another generalized co-letterplace ideal by a regular sequence of variable differences. Finally, we provide a new class of simplicial spheres

Theorems & Definitions (43)

• Definition 1.1
• Theorem 1.3
• Definition 2.1
• Lemma 2.2
• Definition 2.3
• Definition 2.4
• Definition 3.1
• Lemma 3.2
• proof
• Lemma 3.3