Collectanea Mathematica

, Volume 67, Issue 3, pp 357–362 | Cite as

Linear syzygies, flag complexes, and regularity

  • Alexandru Constantinescu
  • Thomas Kahle
  • Matteo Varbaro
Article
  • 126 Downloads

Abstract

We show that for every \(r\in \mathbb {Z}_{>0}\) there exist monomial ideals generated in degree two, with linear syzygies, and regularity of the quotient equal to \(r\). For Gorenstein ideals we prove that the regularity of their quotients can not exceed four, thus showing that for \(d > 4\) every triangulation of a \(d\)-manifold has a hollow square or simplex. We also show that for most monomial ideals generated in degree two and with linear syzygies the regularity is \(O(\log (\log (n))\), where \(n\) is the number of variables.

Mathematics Subject Classification

Primary 13F55 Secondary 13D02 20F55 

References

  1. 1.
    Avramov, L.L., Conca, A., Iyengar, S.B.: Subadditivity of syzygies of Koszul algebras. Math. Ann. 361(1–2), 511–534 (2013)Google Scholar
  2. 2.
    Bayer, D., Mumford, D.: What can be computed in algebraic geometry? In: Computational algebraic geometry and commutative algebra, Sympos. Math., vol. XXXIV. Cambridge University Press, Cortona, pp. 1–48 (1993)Google Scholar
  3. 3.
    Bayer, D., Stillman, M.: On the complexity of computing syzygies. J. Symb. Comput. 6(2), 135–147 (1988)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Davis, M.: The geometry and topology of Coxeter groups, vol. 32. Princeton University Press, Princeton, NJ (2008)Google Scholar
  5. 5.
    Dao, H., Huneke, C., Schweig, J.: Bounds on the regularity and projective dimension of ideals associated to graphs. J. Algebr. Comb. 38(1), 37–55 (2013)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Eisenbud, D., Goto, S.: Linear free resolutions and minimal multiplicity. J. Algebr. 88(1), 89–133 (1984)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Januszkiewicz, T., Świątkowski, J.: Hyperbolic Coxeter groups of large dimension. Comment. Math. Helvetici 78(3), 555–583 (2003)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Mayr, E.W., Meyer, A.R.: The complexity of the word problems for commutative semigroups and polynomial ideals. Adv. Math. 46(3), 305–329 (1982)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Miller, E., Sturmfels, B.: Combinatorial commutative algebra, GTM, vol. 227. Springer, Berlin (2005)MATHGoogle Scholar
  10. 10.
    Stanley, R.P.: Cohen-Macaulay complexes. In: Higher Combinatorics, 31. pp. 51–62 (1977)Google Scholar

Copyright information

© Universitat de Barcelona 2015

Authors and Affiliations

  • Alexandru Constantinescu
    • 1
  • Thomas Kahle
    • 2
  • Matteo Varbaro
    • 3
  1. 1.Mathematisches InstitutFreie Universität BerlinBerlinGermany
  2. 2.Fakultät für MathematikOvGU MagdeburgMagdeburgGermany
  3. 3.Dipartimento di MatematicaUniversità di GenovaGenoaItaly

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