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On the growth of deviations


Authors: Adam Boocher, Alessio D’Alì, Eloísa Grifo, Jonathan Montaño and Alessio Sammartano
Journal: Proc. Amer. Math. Soc. 144 (2016), 5049-5060
MSC (2010): Primary 13D02; Secondary 16E45, 13D40, 16S37, 05C25, 05C38
DOI: https://doi.org/10.1090/proc/13132
Published electronically: August 18, 2016
MathSciNet review: 3556251
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Abstract: The deviations of a graded algebra are a sequence of integers that determine the Poincaré series of its residue field and arise as the number of generators of certain DG algebras. In a sense, deviations measure how far a ring is from being a complete intersection. In this paper, we study extremal deviations among those of algebras with a fixed Hilbert series. In this setting, we prove that, like the Betti numbers, deviations do not increase when passing to an initial ideal and are maximized by the lex-segment ideal. We also prove that deviations grow exponentially for Golod rings and for certain quadratic monomial algebras.


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  • [1] Saeid Alikhani and Yee-hock Peng, Independence roots and independence fractals of certain graphs, J. Appl. Math. Comput. 36 (2011), no. 1-2, 89-100. MR 2794133, https://doi.org/10.1007/s12190-010-0389-4
  • [2] E. F. Assmus Jr., On the homology of local rings, Illinois J. Math. 3 (1959), 187-199. MR 0103907
  • [3] L. L. Avramov, Flat morphisms of complete intersections, Dokl. Akad. Nauk SSSR 225 (1975), no. 1, 11-14 (Russian). MR 0396558
  • [4] Luchezar L. Avramov, Local algebra and rational homotopy, Algebraic homotopy and local algebra (Luminy, 1982) Astérisque, vol. 113, Soc. Math. France, Paris, 1984, pp. 15-43. MR 749041
  • [5] Luchezar L. Avramov, Infinite free resolutions, Six lectures on commutative algebra (Bellaterra, 1996) Progr. Math., vol. 166, Birkhäuser, Basel, 1998, pp. 1-118. MR 1648664
  • [6] Luchezar L. Avramov, Aldo Conca, and Srikanth B. Iyengar, Free resolutions over commutative Koszul algebras, Math. Res. Lett. 17 (2010), no. 2, 197-210. MR 2644369, https://doi.org/10.4310/MRL.2010.v17.n2.a1
  • [7] I. K. Babenko, Analytic properties of Poincaré series of a loop space, Mat. Zametki 27 (1980), no. 5, 751-765, 830 (Russian). MR 578259
  • [8] Jörgen Backelin, Les anneaux locaux à relations monomiales ont des séries de Poincaré-Betti rationnelles, C. R. Acad. Sci. Paris Sér. I Math. 295 (1982), no. 11, 607-610 (French, with English summary). MR 686351
  • [9] Jörgen Backelin and Ralf Fröberg, Koszul algebras, Veronese subrings and rings with linear resolutions, Rev. Roumaine Math. Pures Appl. 30 (1985), no. 2, 85-97. MR 789425
  • [10] Anna Maria Bigatti, Upper bounds for the Betti numbers of a given Hilbert function, Comm. Algebra 21 (1993), no. 7, 2317-2334. MR 1218500, https://doi.org/10.1080/00927879308824679
  • [11] Adam Boocher, Alessio D'Alì, Eloísa Grifo, Jonathan Montaño, and Alessio Sammartano, Edge ideals and DG algebra resolutions, Matematiche (Catania) 70 (2015), no. 1, 215-238. MR 3351467
  • [12] Maria Chudnovsky and Paul Seymour, The roots of the independence polynomial of a clawfree graph, J. Combin. Theory Ser. B 97 (2007), no. 3, 350-357. MR 2305888, https://doi.org/10.1016/j.jctb.2006.06.001
  • [13] Aldo Conca, Koszul algebras and their syzygies, Combinatorial algebraic geometry, Lecture Notes in Math., vol. 2108, Springer, Cham, 2014, pp. 1-31. MR 3329085, https://doi.org/10.1007/978-3-319-04870-3_1
  • [14] Péter Csikvári, Note on the smallest root of the independence polynomial, Combin. Probab. Comput. 22 (2013), no. 1, 1-8. MR 3002570, https://doi.org/10.1017/S0963548312000302
  • [15] Y. Félix and J.-C. Thomas, The radius of convergence of Poincaré series of loop spaces, Invent. Math. 68 (1982), no. 2, 257-274. MR 666163, https://doi.org/10.1007/BF01394059
  • [16] Ralph Fröberg, Determination of a class of Poincaré series, Math. Scand. 37 (1975), no. 1, 29-39. MR 0404254
  • [17] Ralf Fröberg, On Stanley-Reisner rings, Topics in algebra, Part 2 (Warsaw, 1988) Banach Center Publ., vol. 26, PWN, Warsaw, 1990, pp. 57-70. MR 1171260
  • [18] Massimiliano Goldwurm and Massimo Santini, Clique polynomials have a unique root of smallest modulus, Inform. Process. Lett. 75 (2000), no. 3, 127-132. MR 1776664, https://doi.org/10.1016/S0020-0190(00)00086-7
  • [19] D. R. Grayson, M. E. Stillman, Macaulay2, a software system for research in algebraic geometry, available at www.math.uiuc.edu/Macaulay2/.
  • [20] Tor Holtedahl Gulliksen, A proof of the existence of minimal $ R$-algebra resolutions, Acta Math. 120 (1968), 53-58. MR 0224607
  • [21] Stephen Halperin, The nonvanishing of the deviations of a local ring, Comment. Math. Helv. 62 (1987), no. 4, 646-653. MR 920063, https://doi.org/10.1007/BF02564468
  • [22] Jürgen Herzog and Takayuki Hibi, Monomial ideals, Graduate Texts in Mathematics, vol. 260, Springer-Verlag London, Ltd., London, 2011. MR 2724673
  • [23] J. Herzog, V. Reiner, and V. Welker, Componentwise linear ideals and Golod rings, Michigan Math. J. 46 (1999), no. 2, 211-223. MR 1704158, https://doi.org/10.1307/mmj/1030132406
  • [24] Heather A. Hulett, Maximum Betti numbers of homogeneous ideals with a given Hilbert function, Comm. Algebra 21 (1993), no. 7, 2335-2350. MR 1218501, https://doi.org/10.1080/00927879308824680
  • [25] J. Mccullough, I. Peeva, Infinite graded free resolutions, to appear in Commutative Algebra and Noncommutative Algebraic Geometry (Eisenbud, Iyengar, Singh, Stafford, Van den Bergh, Eds.), Math. Sci. Res. Inst. Publ., Cambridge University Press.
  • [26] Keith Pardue, Deformation classes of graded modules and maximal Betti numbers, Illinois J. Math. 40 (1996), no. 4, 564-585. MR 1415019
  • [27] Irena Peeva, 0-Borel fixed ideals, J. Algebra 184 (1996), no. 3, 945-984. MR 1407879, https://doi.org/10.1006/jabr.1996.0293
  • [28] Irena Peeva, Consecutive cancellations in Betti numbers, Proc. Amer. Math. Soc. 132 (2004), no. 12, 3503-3507. MR 2084070, https://doi.org/10.1090/S0002-9939-04-07517-3
  • [29] Alexander Polishchuk and Leonid Positselski, Quadratic algebras, University Lecture Series, vol. 37, American Mathematical Society, Providence, RI, 2005. MR 2177131
  • [30] Jan-Erik Roos, On computer-assisted research in homological algebra, Math. Comput. Simulation 42 (1996), no. 4-6, 475-490. Symbolic computation, new trends and developments (Lille, 1993). MR 1430835, https://doi.org/10.1016/S0378-4754(96)00023-7
  • [31] Colette Schoeller, Homologie des anneaux locaux noethériens, C. R. Acad. Sci. Paris Sér. A-B 265 (1967), A768-A771 (French). MR 0224682
  • [32] Li-Chuan Sun, Growth of Betti numbers of modules over local rings of small embedding codimension or small linkage number, J. Pure Appl. Algebra 96 (1994), no. 1, 57-71. MR 1297441, https://doi.org/10.1016/0022-4049(94)90087-6
  • [33] John Tate, Homology of Noetherian rings and local rings, Illinois J. Math. 1 (1957), 14-27. MR 0086072
  • [34] Jan Uliczka, Remarks on Hilbert series of graded modules over polynomial rings, Manuscripta Math. 132 (2010), no. 1-2, 159-168. MR 2609292, https://doi.org/10.1007/s00229-010-0341-9

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Additional Information

Adam Boocher
Affiliation: School of Mathematics, University of Edinburgh, James Clerk Maxwell Building, Mayfield Road, Edinburgh EH9 3JZ, Scotland
Email: adam.boocher@ed.ac.uk

Alessio D’Alì
Affiliation: Dipartimento di Matematica, Università degli Studi di Genova, Via Dodecaneso 35, 16146 Genova, Italy
Email: dali@dima.unige.it

Eloísa Grifo
Affiliation: Department of Mathematics, University of Virginia, 141 Cabell Drive, Kerchof Hall, Charlottesville, Virginia 22904
Email: er2eq@virginia.edu

Jonathan Montaño
Affiliation: Department of Mathematics, University of Kansas, 405 Snow Hall, 1460 Jayhawk Boulevard, Lawrence, Kansas 66045
Email: jmontano@ku.edu

Alessio Sammartano
Affiliation: Department of Mathematics, Purdue University, 150 North University Street, West Lafayette, Indiana 47907
Email: asammart@purdue.edu

DOI: https://doi.org/10.1090/proc/13132
Received by editor(s): March 30, 2015
Received by editor(s) in revised form: January 27, 2016
Published electronically: August 18, 2016
Communicated by: Irena Peeva
Article copyright: © Copyright 2016 American Mathematical Society

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