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Proceedings of the American Mathematical Society

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Average behavior of minimal free resolutions of monomial ideals


Authors: Jesús A. De Loera, Serkan Hoşten, Robert Krone and Lily Silverstein
Journal: Proc. Amer. Math. Soc. 147 (2019), 3239-3257
MSC (2010): Primary 13D02, 13P20
DOI: https://doi.org/10.1090/proc/14403
Published electronically: April 18, 2019
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Abstract: We show that, under a natural probability distribution, random monomial ideals will almost always have minimal free resolutions of maximal length; that is, the projective dimension will almost always be $ n$, where $ n$ is the number of variables in the polynomial ring. As a consequence we prove that Cohen-Macaulayness is a rare property. We characterize when a random monomial ideal is generic/strongly generic, and when it is Scarf--i.e., when the algebraic Scarf complex of $ M\subset S=k[x_1,\ldots ,x_n]$ gives a minimal free resolution of $ S/M$. It turns out, outside of a very specific ratio of model parameters, random monomial ideals are Scarf only when they are generic. We end with a discussion of the average magnitude of Betti numbers.


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Additional Information

Jesús A. De Loera
Affiliation: Department of Mathematics, University of California, Davis, California 95616
Email: deloera@math.ucdavis.edu

Serkan Hoşten
Affiliation: Department of Mathematics, San Francisco State University, San Francisco, California 94132
Email: serkan@sfsu.edu

Robert Krone
Affiliation: Department of Mathematics, University of California, Davis, California 95616
Email: rckrone@ucdavis.edu

Lily Silverstein
Affiliation: Department of Mathematics, University of California, Davis, California 95616
Email: lsilver@math.ucdavis.edu

DOI: https://doi.org/10.1090/proc/14403
Received by editor(s): March 23, 2018
Received by editor(s) in revised form: July 24, 2018, and September 17, 2018
Published electronically: April 18, 2019
Additional Notes: This work was conducted and prepared at the Mathematical Sciences Research Institute in Berkeley, California, during the fall 2017 semester. Thus we gratefully acknowledge partial support by NSF grant DMS-1440140.
In addition, the first and fourth authors were also partially supported by NSF grant DMS-1522158.
Computer simulations made use of the Random Monomial Ideals package \cite{RMIpackage} for Macaulay2 \cite{M2}.
Communicated by: Claudia Polini
Article copyright: © Copyright 2019 American Mathematical Society