On the generalized Scarf complex of lattice ideals

TL;DR

This paper introduces the generalized Scarf complex for lattice ideals, demonstrating its essential role in all minimal free resolutions, thus advancing the understanding of lattice ideal structures.

Contribution

It defines the generalized Scarf complex for lattice ideals and proves its indispensability in minimal free resolutions, a novel insight in algebraic combinatorics.

Findings

The generalized Scarf complex is contained in every minimal free resolution.

The paper establishes the importance of the generalized Scarf complex in lattice ideal theory.

It provides a new tool for analyzing the structure of lattice ideals.

Abstract

Let $k$ be a field, $ \mathcal{L}\subset \mathbb{Z}^n$ be a lattice such that $\L\cap \mathbb{N}^n=\{{\bf 0}\}$, and $I_\L\subset \Bbbk[x_1,..., x_n]$ the corresponding lattice ideal. We present the generalized Scarf complex of $I_\L$ and show that it is indispensable in the sense that it is contained in every minimal free resolution of $R/I_\L$.