Minimal free resolution of Crystal module

TL;DR

This paper investigates the minimal free resolution of the Crystal module associated with the join-meet ideal of non-distributive finite lattices, specifically focusing on the Crystal lattice, and provides inequalities for projective dimension and Betti numbers.

Contribution

It introduces inequalities for the minimal free resolution of the Crystal module, linking projective dimension and Betti numbers to the Koszul complex.

Findings

Projective dimension of the Crystal module can be evaluated via the Koszul complex.

Betti numbers of the Crystal module are determined using the Koszul complex.

Provides bounds for the minimal free resolution of the Crystal module.

Abstract

The join-meet ideal was introduced by Takayuki Hibi in 1987. It is binomial ideals that are defined by finite lattices. We study the join-meet ideal of non-distributive finite lattices that do not always satisfy modular. In particular, we work on the case of Crystal lattice which is one of them. It was introduced by Yohei Oshida in 2022. The Crystal module in the title is the residue class ring which is in relation to the join-meet ideal of Crystal lattice. In this paper, we give important inequalities about the minimal free resolution of the Crystal module. The important point about this result is that the projective dimension and Betti-number of the Crystal module can be evaluated by using the Koszul complex.