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The join-meet ideal was introduced by Takayuki Hibi in 1987. It is binomialideals that are defined by finite lattices. We study the join-meet ideal ofnon-distributive finite lattices that do not always satisfy modular. Inparticular, we work on the case of Crystal lattice which is one of them. It wasintroduced by Yohei Oshida in 2022. The Crystal module in the title is theresidue class ring which is in relation to the join-meet ideal of Crystallattice. In this paper, we give important inequalities about the minimal freeresolution of the Crystal module. The important point about this result is thatthe projective dimension and Betti-number of the Crystal module can beevaluated by using the Koszul complex.

Minimal free resolution of Crystal module