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
MathSciNet review:
3981105
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Abstract | References | Similar Articles | Additional Information
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
MR Author ID:
364032
ORCID:
0000-0002-9556-1112
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
MR Author ID:
1022926
Email:
rckrone@ucdavis.edu
Lily Silverstein
Affiliation:
Department of Mathematics, University of California, Davis, California 95616
ORCID:
0000-0003-4368-9912
Email:
lsilver@math.ucdavis.edu
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