On the growth of deviations
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- by Adam Boocher, Alessio D’Alì, Eloísa Grifo, Jonathan Montaño and Alessio Sammartano PDF
- Proc. Amer. Math. Soc. 144 (2016), 5049-5060 Request permission
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.References
- 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, DOI 10.1007/s12190-010-0389-4
- E. F. Assmus Jr., On the homology of local rings, Illinois J. Math. 3 (1959), 187–199. MR 103907
- L. L. Avramov, Flat morphisms of complete intersections, Dokl. Akad. Nauk SSSR 225 (1975), no. 1, 11–14 (Russian). MR 0396558
- 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
- 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
- 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, DOI 10.4310/MRL.2010.v17.n2.a1
- I. K. Babenko, Analytic properties of Poincaré series of a loop space, Mat. Zametki 27 (1980), no. 5, 751–765, 830 (Russian). MR 578259
- 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
- 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
- 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, 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, DOI 10.4418/2015.70.1.16
- 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, DOI 10.1016/j.jctb.2006.06.001
- Aldo Conca, Koszul algebras and their syzygies, Combinatorial algebraic geometry, Lecture Notes in Math., vol. 2108, Springer, Cham, 2014, pp. 1–31. MR 3329085, DOI 10.1007/978-3-319-04870-3_{1}
- Péter Csikvári, Note on the smallest root of the independence polynomial, Combin. Probab. Comput. 22 (2013), no. 1, 1–8. MR 3002570, DOI 10.1017/S0963548312000302
- 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, DOI 10.1007/BF01394059
- Ralph Fröberg, Determination of a class of Poincaré series, Math. Scand. 37 (1975), no. 1, 29–39. MR 404254, DOI 10.7146/math.scand.a-11585
- 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
- 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, DOI 10.1016/S0020-0190(00)00086-7
- D. R. Grayson, M. E. Stillman, Macaulay2, a software system for research in algebraic geometry, available at www.math.uiuc.edu/Macaulay2/.
- Tor Holtedahl Gulliksen, A proof of the existence of minimal $R$-algebra resolutions, Acta Math. 120 (1968), 53–58. MR 224607, DOI 10.1007/BF02394606
- Stephen Halperin, The nonvanishing of the deviations of a local ring, Comment. Math. Helv. 62 (1987), no. 4, 646–653. MR 920063, DOI 10.1007/BF02564468
- 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
- J. Herzog, V. Reiner, and V. Welker, Componentwise linear ideals and Golod rings, Michigan Math. J. 46 (1999), no. 2, 211–223. MR 1704158, DOI 10.1307/mmj/1030132406
- 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
- 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.
- Keith Pardue, Deformation classes of graded modules and maximal Betti numbers, Illinois J. Math. 40 (1996), no. 4, 564–585. MR 1415019
- Irena Peeva, $0$-Borel fixed ideals, J. Algebra 184 (1996), no. 3, 945–984. MR 1407879, DOI 10.1006/jabr.1996.0293
- Irena Peeva, Consecutive cancellations in Betti numbers, Proc. Amer. Math. Soc. 132 (2004), no. 12, 3503–3507. MR 2084070, DOI 10.1090/S0002-9939-04-07517-3
- Alexander Polishchuk and Leonid Positselski, Quadratic algebras, University Lecture Series, vol. 37, American Mathematical Society, Providence, RI, 2005. MR 2177131, DOI 10.1090/ulect/037
- 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, DOI 10.1016/S0378-4754(96)00023-7
- Colette Schoeller, Homologie des anneaux locaux noethériens, C. R. Acad. Sci. Paris Sér. A-B 265 (1967), A768–A771 (French). MR 224682
- 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, DOI 10.1016/0022-4049(94)90087-6
- John Tate, Homology of Noetherian rings and local rings, Illinois J. Math. 1 (1957), 14–27. MR 86072
- Jan Uliczka, Remarks on Hilbert series of graded modules over polynomial rings, Manuscripta Math. 132 (2010), no. 1-2, 159–168. MR 2609292, DOI 10.1007/s00229-010-0341-9
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
- MR Author ID: 1111831
- 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
- MR Author ID: 890186
- Email: jmontano@ku.edu
- Alessio Sammartano
- Affiliation: Department of Mathematics, Purdue University, 150 North University Street, West Lafayette, Indiana 47907
- MR Author ID: 942872
- ORCID: 0000-0002-0377-1375
- Email: asammart@purdue.edu
- 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
- © Copyright 2016 American Mathematical Society
- 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
- MathSciNet review: 3556251