Average behavior of minimal free resolutions of monomial ideals
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- by Jesús A. De Loera, Serkan Hoşten, Robert Krone and Lily Silverstein PDF
- Proc. Amer. Math. Soc. 147 (2019), 3239-3257 Request permission
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.References
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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
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 3239-3257
- MSC (2010): Primary 13D02, 13P20
- DOI: https://doi.org/10.1090/proc/14403
- MathSciNet review: 3981105