Monomial curves and locally linear resolution

TL;DR

The paper addresses whether locally linear graded free resolutions with arbitrarily large gaps exist. It develops an explicit construction of complexes of free $A$-modules from four elements $f_1,\dots,f_4$, derives exactness criteria via grade and regularity, and uses these to obtain detailed resolutions and dualities for associated modules. A key application computes the Hartshorne–Rao module $HR(C)$ of a monomial curve on a smooth quadric, yielding an explicit minimal free resolution and showing that a locally linear resolution with an unbounded gap occurs in this geometric setting. Overall, the work provides a concrete framework for producing locally linear resolutions with large gaps and demonstrates this phenomenon in the context of monomial curves and Hartshorne–Rao theory, including linearly generated modules with high-degree syzygies.

Abstract

For four elements of a Noetherian ring we construct complexes of free modules of length three (resp. five) by an explicit description of the homomorphisms of the free modules. We provide exactness criteria for them. As an application we use these results in order to describe explicit the minimal free resolution of the Hartshorne--Rao module of a monomial curve lying on a smooth quadric. Also it provides an example of linearly generated module with syzygies of arbitrary high degree.

Theorems & Definitions (24)

• Theorem 1.2
• Corollary 1.3
• Remark 2.3
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Theorem 4.1
• proof
• Theorem 4.2