RIGID MONOMIAL IDEALS

In this paper we investigate the class of rigid monomial ideals and characterize them by the fact that their minimal resolution has a unique Zd-graded basis. Furthermore, we show that certain rigid monomial ideals are lattice-linear, so their minimal resolution can be constructed as a poset resoluti...

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Bibliographic Details
Published in:Journal of commutative algebra 2014-03, Vol.6 (1), p.33-52
Main Authors: CLARK, TIMOTHY B.P., MAPES, SONJA
Format: Article
Language:English
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Summary:In this paper we investigate the class of rigid monomial ideals and characterize them by the fact that their minimal resolution has a unique Zd-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 L(n), the lattice of all lcm-lattices of monomial ideals on n generators. By fixing a stratum in L(n) where all ideals have the same total Betti numbers, we show that rigidity is a property which propagates upward in 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.
ISSN:1939-0807
1939-2346
1939-2346
DOI:10.1216/JCA-2014-6-1-33