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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Minimal free resolutions of monomial ideals and of toric rings are supported on posets


Authors: Timothy B. P. Clark and Alexandre B. Tchernev
Journal: Trans. Amer. Math. Soc. 371 (2019), 3995-4027
MSC (2010): Primary 13D02, 06A11, 14M25; Secondary 05E40
DOI: https://doi.org/10.1090/tran/7614
Published electronically: October 23, 2018
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Abstract: We introduce the notion of a resolution supported on a poset. When the poset is a CW-poset, i.e., the face poset of a regular CW-complex, we recover the notion of cellular resolution as introduced by Bayer and Sturmfels. Work of Reiner and Welker, and of Velasco, has shown that there are monomial ideals whose minimal free resolutions are not cellular, hence cannot be supported on any CW-poset. We show that for any monomial ideal there is a homology CW-poset that supports a minimal free resolution of the ideal. This allows one to extend to every minimal resolution, essentially verbatim, techniques initially developed to study cellular resolutions. As two demonstrations of this process, we show that minimal resolutions of toric rings are supported on what we call toric hcw-posets, and, generalizing results of Miller and Sturmfels, we prove a fundamental relationship between Artinianizations and Alexander duality for monomial ideals.


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Additional Information

Timothy B. P. Clark
Affiliation: Department of Mathematics and Statistics, Loyola University Maryland, 4501 North Charles Street, Baltimore, Maryland 21210
Email: tbclark@loyola.edu

Alexandre B. Tchernev
Affiliation: Department of Mathematics and Statistics, University at Albany, SUNY, 1400 Washington Avenue, Albany, New York 12222
Email: atchernev@albany.edu

DOI: https://doi.org/10.1090/tran/7614
Received by editor(s): October 9, 2015
Received by editor(s) in revised form: July 14, 2017, and November 9, 2017
Published electronically: October 23, 2018
Article copyright: © Copyright 2018 American Mathematical Society