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Asymptotic syzygies of Stanley-Reisner rings of iterated subdivisions


Authors: Aldo Conca, Martina Juhnke-Kubitzke and Volkmar Welker
Journal: Trans. Amer. Math. Soc. 370 (2018), 1661-1691
MSC (2010): Primary 13F55, 05E45
DOI: https://doi.org/10.1090/tran/7149
Published electronically: September 7, 2017
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Abstract: Inspired by recent results of Ein, Lazarsfeld, Erman and Zhou on the non-vanishing of Betti numbers of high Veronese subrings, we describe the behavior of the Betti numbers of Stanley-Reisner rings associated with iterated barycentric or edgewise subdivisions of a given simplicial complex. Our results show that for a simplicial complex $ \Delta $ of dimension $ d-1$ and for $ 1\leq j\leq d-1$ the number of 0's in the $ j^{\text {th}}$ linear strand of the minimal free resolution of the $ r^{\text {th}}$ barycentric or edgewise subdivision is bounded above only in terms of $ d$ and $ j$ (and independently of $ r$).


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  • [1] Luchezar L. Avramov, Aldo Conca, and Srikanth B. Iyengar, Subadditivity of syzygies of Koszul algebras, Math. Ann. 361 (2015), no. 1-2, 511-534. MR 3302628, https://doi.org/10.1007/s00208-014-1060-4
  • [2] Matthias Beck and Alan Stapledon, On the log-concavity of Hilbert series of Veronese subrings and Ehrhart series, Math. Z. 264 (2010), no. 1, 195-207. MR 2564938, https://doi.org/10.1007/s00209-008-0458-7
  • [3] A. Björner, Topological methods, Handbook of combinatorics, Vol. 1, 2, Elsevier Sci. B. V., Amsterdam, 1995, pp. 1819-1872. MR 1373690
  • [4] Anders Björner, Michel Las Vergnas, Bernd Sturmfels, Neil White, and Günter M. Ziegler, Oriented matroids, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 46, Cambridge University Press, Cambridge, 1999. MR 1744046
  • [5] A. Björner and S. Linusson, The number of $ k$-faces of a simple $ d$-polytope, Discrete Comput. Geom. 21 (1999), no. 1, 1-16. MR 1661283, https://doi.org/10.1007/PL00009403
  • [6] Francesco Brenti and Volkmar Welker, $ f$-vectors of barycentric subdivisions, Math. Z. 259 (2008), no. 4, 849-865. MR 2403744, https://doi.org/10.1007/s00209-007-0251-z
  • [7] Francesco Brenti and Volkmar Welker, The Veronese construction for formal power series and graded algebras, Adv. in Appl. Math. 42 (2009), no. 4, 545-556. MR 2511015, https://doi.org/10.1016/j.aam.2009.01.001
  • [8] Winfried Bruns, Aldo Conca, and Tim Römer, Koszul homology and syzygies of Veronese subalgebras, Math. Ann. 351 (2011), no. 4, 761-779. MR 2854112, https://doi.org/10.1007/s00208-010-0616-1
  • [9] Morten Brun and Tim Römer, Subdivisions of toric complexes, J. Algebraic Combin. 21 (2005), no. 4, 423-448. MR 2153934, https://doi.org/10.1007/s10801-005-3020-2
  • [10] Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR 1251956
  • [11] Marc Chardin, Jean-Pierre Jouanolou, and Ahad Rahimi, The eventual stability of depth, associated primes and cohomology of a graded module, J. Commut. Algebra 5 (2013), no. 1, 63-92. MR 3084122
  • [12] Jeff Cheeger, Werner Müller, and Robert Schrader, On the curvature of piecewise flat spaces, Comm. Math. Phys. 92 (1984), no. 3, 405-454. MR 734226
  • [13] Hailong Dao, Craig Huneke, and Jay Schweig, Bounds on the regularity and projective dimension of ideals associated to graphs, J. Algebraic Combin. 38 (2013), no. 1, 37-55. MR 3070118, https://doi.org/10.1007/s10801-012-0391-z
  • [14] Emanuele Delucchi, Aaron Pixton, and Lucas Sabalka, Face vectors of subdivided simplicial complexes, Discrete Math. 312 (2012), no. 2, 248-257. MR 2852583, https://doi.org/10.1016/j.disc.2011.08.032
  • [15] Persi Diaconis and Jason Fulman, Carries, shuffling, and symmetric functions, Adv. in Appl. Math. 43 (2009), no. 2, 176-196. MR 2531920, https://doi.org/10.1016/j.aam.2009.02.002
  • [16] Lawrence Ein and Robert Lazarsfeld, Syzygies and Koszul cohomology of smooth projective varieties of arbitrary dimension, Invent. Math. 111 (1993), no. 1, 51-67. MR 1193597, https://doi.org/10.1007/BF01231279
  • [17] Lawrence Ein and Robert Lazarsfeld, Asymptotic syzygies of algebraic varieties, Invent. Math. 190 (2012), no. 3, 603-646. MR 2995182, https://doi.org/10.1007/s00222-012-0384-5
  • [18] Lawrence Ein, Daniel Erman, and Robert Lazarsfeld, A quick proof of nonvanishing for asymptotic syzygies, Algebr. Geom. 3 (2016), no. 2, 211-222. MR 3477954, https://doi.org/10.14231/AG-2016-010
  • [19] David Eisenbud, Commutative algebra: With a view toward algebraic geometry, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. MR 1322960
  • [20] Światosław R. Gal, Real root conjecture fails for five- and higher-dimensional spheres, Discrete Comput. Geom. 34 (2005), no. 2, 269-284. MR 2155722, https://doi.org/10.1007/s00454-005-1171-5
  • [21] Shiro Goto and Keiichi Watanabe, On graded rings. I, J. Math. Soc. Japan 30 (1978), no. 2, 179-213. MR 494707, https://doi.org/10.2969/jmsj/03020179
  • [22] Martina Kubitzke and Volkmar Welker, The multiplicity conjecture for barycentric subdivisions, Comm. Algebra 36 (2008), no. 11, 4223-4248. MR 2460412, https://doi.org/10.1080/00927870802177333
  • [23] Carl William Lee, Counting the faces of simplicial convex polytopes, ProQuest LLC, Ann Arbor, MI. Thesis (Ph.D.)-Cornell University, 1981. MR 2631156
  • [24] James R. Munkres, Elements of algebraic topology, Addison-Wesley Publishing Company, Menlo Park, CA, 1984. MR 755006
  • [25] James R. Munkres, Topological results in combinatorics, Michigan Math. J. 31 (1984), no. 1, 113-128. MR 736476, https://doi.org/10.1307/mmj/1029002969
  • [26] Richard P. Stanley, Subdivisions and local $ h$-vectors, J. Amer. Math. Soc. 5 (1992), no. 4, 805-851. MR 1157293, https://doi.org/10.2307/2152711
  • [27] James W. Walker, Canonical homeomorphisms of posets, European J. Combin. 9 (1988), no. 2, 97-107. MR 939858, https://doi.org/10.1016/S0195-6698(88)80033-7
  • [28] Xin Zhou, Effective non-vanishing of asymptotic adjoint syzygies, Proc. Amer. Math. Soc. 142 (2014), no. 7, 2255-2264. MR 3195751, https://doi.org/10.1090/S0002-9939-2014-11947-2

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Additional Information

Aldo Conca
Affiliation: DIMA – Dipartimento di Matematica, University of Genova, Via Dodecanesco 35, 16146 Genova, Italy
Email: conca@dima.unige.it

Martina Juhnke-Kubitzke
Affiliation: Fachbereich Informatik und Mathematik, Goethe-Universität Frankfurt, Postfach 11 19 32, D-60054 Frankfurt am Main, Germany
Address at time of publication: Institut für Mathematik, Universität Osnabrück, Albrechtstraße 28a, 49076 Osnabrück, Germany
Email: juhnke-kubitzke@uos.de

Volkmar Welker
Affiliation: Fachbereich Mathematik und Informatik, Philipps-Universität Marburg, Hans- Meerwein-Straße 6, 35032 Marburg, Germany
Email: welker@mathematik.uni-marburg.de

DOI: https://doi.org/10.1090/tran/7149
Keywords: Betti numbers, subdivision, Stanley-Reisner ring
Received by editor(s): November 30, 2015
Received by editor(s) in revised form: May 23, 2016
Published electronically: September 7, 2017
Article copyright: © Copyright 2017 American Mathematical Society

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