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SPRING 2014 Rigid monomial ideals
Timothy B.P. Clark, Sonja Mapes
J. Commut. Algebra 6(1): 33-51 (SPRING 2014). DOI: 10.1216/JCA-2014-6-1-33


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.


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Timothy B.P. Clark. Sonja Mapes. "Rigid monomial ideals." J. Commut. Algebra 6 (1) 33 - 51, SPRING 2014.


Published: SPRING 2014
First available in Project Euclid: 2 June 2014

zbMATH: 1295.13020
MathSciNet: MR3215560
Digital Object Identifier: 10.1216/JCA-2014-6-1-33

Rights: Copyright © 2014 Rocky Mountain Mathematics Consortium


Vol.6 • No. 1 • SPRING 2014
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