Average behavior of minimal free resolutions of monomial ideals
HTML articles powered by AMS MathViewer
- by Jesús A. De Loera, Serkan Hoşten, Robert Krone and Lily Silverstein
- Proc. Amer. Math. Soc. 147 (2019), 3239-3257
- DOI: https://doi.org/10.1090/proc/14403
- Published electronically: April 18, 2019
- PDF | 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
- Guillermo Alesandroni, Minimal resolutions of dominant and semidominant ideals, J. Pure Appl. Algebra 221 (2017), no. 4, 780–798. MR 3574207, DOI 10.1016/j.jpaa.2016.08.003
- Guillermo Alesandroni, Monomial ideals with large projective dimension, arXiv preprint arXiv:1710.05124 (2017).
- Dave Bayer, Irena Peeva, and Bernd Sturmfels, Monomial resolutions, Math. Res. Lett. 5 (1998), no. 1-2, 31–46. MR 1618363, DOI 10.4310/MRL.1998.v5.n1.a3
- Anna Maria Bigatti, Upper bounds for the Betti numbers of a given Hilbert function, Comm. Algebra 21 (1993), no. 7, 2317–2334. MR 1218500, DOI 10.1080/00927879308824679
- Adam Boocher and James Seiner, Lower bounds for Betti numbers of monomial ideals, J. Algebra 508 (2018), 445–460. MR 3810302, DOI 10.1016/j.jalgebra.2018.04.013
- Morten Brun and Tim Römer, Betti numbers of ${\Bbb Z}^n$-graded modules, Comm. Algebra 32 (2004), no. 12, 4589–4599. MR 2111102, DOI 10.1081/AGB-200036803
- David A. Buchsbaum and David Eisenbud, Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension $3$, Amer. J. Math. 99 (1977), no. 3, 447–485. MR 453723, DOI 10.2307/2373926
- Hara Charalambous, Betti numbers of multigraded modules, J. Algebra 137 (1991), no. 2, 491–500. MR 1094254, DOI 10.1016/0021-8693(91)90103-F
- David Cox, John Little, and Donal O’Shea, Ideals, varieties, and algorithms, 3rd ed., Undergraduate Texts in Mathematics, Springer, New York, 2007. An introduction to computational algebraic geometry and commutative algebra. MR 2290010, DOI 10.1007/978-0-387-35651-8
- Jesús A. De Loera, Sonja Petrovic, Lily Silverstein, Despina Stasi, and Dane Wilburne, Random monomial ideals, to appear in Journal of Algebra, available at arXiv:1701.07130 (2017).
- Lawrence Ein, Daniel Erman, and Robert Lazarsfeld, Asymptotics of random Betti tables, J. Reine Angew. Math. 702 (2015), 55–75. MR 3341466, DOI 10.1515/crelle-2013-0032
- David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960, DOI 10.1007/978-1-4612-5350-1
- David Eisenbud and Frank-Olaf Schreyer, Betti numbers of graded modules and cohomology of vector bundles, J. Amer. Math. Soc. 22 (2009), no. 3, 859–888. MR 2505303, DOI 10.1090/S0894-0347-08-00620-6
- Daniel Erman and Jay Yang, Random flag complexes and asymptotic syzygies, arXiv preprint arXiv:1706.01488 (2017).
- Daniel R. Grayson and Michael E. Stillman, Macaulay2, a software system for research in algebraic geometry, Available at https://faculty.math.illinois.edu/Macaulay2/.
- Robin Hartshorne, Algebraic vector bundles on projective spaces: a problem list, Topology 18 (1979), no. 2, 117–128. MR 544153, DOI 10.1016/0040-9383(79)90030-2
- Jürgen Herzog and Takayuki Hibi, Monomial ideals, Graduate Texts in Mathematics, vol. 260, Springer-Verlag London, Ltd., London, 2011. MR 2724673, DOI 10.1007/978-0-85729-106-6
- Serkan Hoşten and Walter D. Morris Jr., The order dimension of the complete graph, Discrete Math. 201 (1999), no. 1-3, 133–139. MR 1687882, DOI 10.1016/S0012-365X(98)00315-X
- Heather A. Hulett, Maximum Betti numbers of homogeneous ideals with a given Hilbert function, Comm. Algebra 21 (1993), no. 7, 2335–2350. MR 1218501, DOI 10.1080/00927879308824680
- Roberto La Scala and Michael Stillman, Strategies for computing minimal free resolutions, J. Symbolic Comput. 26 (1998), no. 4, 409–431. MR 1646662, DOI 10.1006/jsco.1998.0221
- Ezra Miller and Bernd Sturmfels, Combinatorial commutative algebra, Graduate Texts in Mathematics, vol. 227, Springer-Verlag, New York, 2005. MR 2110098
- Keith Pardue, Deformation classes of graded modules and maximal Betti numbers, Illinois J. Math. 40 (1996), no. 4, 564–585. MR 1415019
- Sonja Petrović, Despina Stasi, and Dane Wilburne, Random Monomial Ideals Macaulay2 Package, ArXiv e-prints (2017).
- Neil J.A. Sloane, The Online Encyclopedia of Integer Sequences, A001206, Available at https://oeis.org/A001206.
- Mark E. Walker, Total Betti numbers of modules of finite projective dimension, Ann. of Math. (2) 186 (2017), no. 2, 641–646. MR 3702675, DOI 10.4007/annals.2017.186.2.6
Bibliographic 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