## Journal of Commutative Algebra

### Rigid monomial ideals

#### Abstract

In this paper we investigate the class of rigid monomial ideals and characterize them by the fact that their minimal resolution has a unique $\mathbf{Z}^d$-graded basis. Furthermore, we show that certain rigid monomial ideals are lattice-linear, so their minimal resolution can be constructed as a poset resolution. We then give a description of the minimal resolution of a larger class of rigid monomial ideals by appealing to the structure of $\mathcal{L}(n)$, the lattice of all lcm-lattices of monomial ideals on $n$ generators. By fixing a stratum in $\mathcal{L}(n)$ where all ideals have the same total Betti numbers, we show that rigidity is a property which propagates upward in $\mathcal{L}(n)$. This allows the minimal resolution of any rigid ideal contained in a fixed stratum to be constructed by relabeling the resolution of a rigid monomial ideal whose resolution has been constructed by other methods.

#### Article information

Source
J. Commut. Algebra, Volume 6, Number 1 (2014), 33-51.

Dates
First available in Project Euclid: 2 June 2014

https://projecteuclid.org/euclid.jca/1401715577

Digital Object Identifier
doi:10.1216/JCA-2014-6-1-33

Mathematical Reviews number (MathSciNet)
MR3215560

Zentralblatt MATH identifier
1295.13020

#### Citation

Clark, Timothy B.P.; Mapes, Sonja. Rigid monomial ideals. J. Commut. Algebra 6 (2014), no. 1, 33--51. doi:10.1216/JCA-2014-6-1-33. https://projecteuclid.org/euclid.jca/1401715577

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