Advertisement

Quasi-stability versus Genericity

  • Amir Hashemi
  • Michael Schweinfurter
  • Werner M. Seiler
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7442)

Abstract

Quasi-stable ideals appear as leading ideals in the theory of Pommaret bases. We show that quasi-stable leading ideals share many of the properties of the generic initial ideal. In contrast to genericity, quasi-stability is a characteristic independent property that can be effectively verified. We also relate Pommaret bases to some invariants associated with local cohomology, exhibit the existence of linear quotients in Pommaret bases and prove some results on componentwise linear ideals.

Keywords

Betti Number Polynomial Ideal Monomial Ideal Free Resolution Minimal Basis 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aramova, A., Herzog, J.: Almost regular sequences and Betti numbers. Amer. J. Math. 122, 689–719 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Aramova, A., Herzog, J., Hibi, T.: Ideals with stable Betti numbers. Adv. Math. 152, 72–77 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bayer, D., Charalambous, H., Popescu, S.: Extremal Betti numbers and applications to monomial ideals. J. Alg. 221, 497–512 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Bayer, D., Stillman, M.: A criterion for detecting m-regularity. Invent. Math. 87, 1–11 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Bayer, D., Stillman, M.: A theorem on refining division orders by the reverse lexicographic orders. Duke J. Math. 55, 321–328 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Bermejo, I., Gimenez, P.: Saturation and Castelnuovo-Mumford regularity. J. Alg. 303, 592–617 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Caviglia, G., Sbarra, E.: Characteristic-free bounds for the Castelnuovo-Mumford regularity. Compos. Math. 141, 1365–1373 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    CoCoATeam: CoCoA: a system for doing Computations in Commutative Algebra, http://cocoa.dima.unige.it
  9. 9.
    Eisenbud, D.: Commutative Algebra with a View Toward Algebraic Geometry. Graduate Texts in Mathematics, vol. 150. Springer, New York (1995)zbMATHGoogle Scholar
  10. 10.
    Galligo, A.: A propos du théorème de préparation de Weierstrass. In: Norguet, F. (ed.) Fonctions de Plusieurs Variables Complexes. Lecture Notes in Mathematics, vol. 409, pp. 543–579. Springer, Berlin (1974)CrossRefGoogle Scholar
  11. 11.
    Galligo, A.: Théorème de division et stabilité en géometrie analytique locale. Ann. Inst. Fourier 29(2), 107–184 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Gerdt, V., Blinkov, Y.: Involutive bases of polynomial ideals. Math. Comp. Simul. 45, 519–542 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Green, M.: Generic initial ideals. In: Elias, J., Giral, J., Miró-Roig, R., Zarzuela, S. (eds.) Six Lectures on Commutative Algebra. Progress in Mathematics, vol. 166, pp. 119–186. Birkhäuser, Basel (1998)Google Scholar
  14. 14.
    Guillemin, V., Sternberg, S.: An algebraic model of transitive differential geometry. Bull. Amer. Math. Soc. 70, 16–47 (1964), (With a letter of Serre as appendix)Google Scholar
  15. 15.
    Hausdorf, M., Sahbi, M., Seiler, W.: δ- and quasi-regularity for polynomial ideals. In: Calmet, J., Seiler, W., Tucker, R. (eds.) Global Integrability of Field Theories, pp. 179–200. Universitätsverlag Karlsruhe, Karlsruhe (2006)Google Scholar
  16. 16.
    Herzog, J., Hibi, T.: Componentwise linear ideals. Nagoya Math. J. 153, 141–153 (1999)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Herzog, J., Hibi, T.: Monomial Ideals. Graduate Texts in Mathematics, vol. 260. Springer, London (2011)zbMATHCrossRefGoogle Scholar
  18. 18.
    Herzog, J., Kühl, M.: On the Bettinumbers of finite pure and linear resolutions. Comm. Alg. 12, 1627–1646 (1984)zbMATHCrossRefGoogle Scholar
  19. 19.
    Herzog, J., Popescu, D., Vladoiu, M.: On the Ext-modules of ideals of Borel type. In: Commutative Algebra. Contemp. Math, vol. 331, pp. 171–186. Amer. Math. Soc., Providence (2003)CrossRefGoogle Scholar
  20. 20.
    Herzog, J., Takayama, Y.: Resolutions by mapping cones. Homol. Homot. Appl. 4, 277–294 (2002)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Mall, D.: On the relation between Gröbner and Pommaret bases. Appl. Alg. Eng. Comm. Comp. 9, 117–123 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Nagel, U., Römer, T.: Criteria for componentwise linearity. Preprint arXiv:1108.3921 (2011)Google Scholar
  23. 23.
    Schenzel, P., Trung, N., Cuong, N.: Verallgemeinerte Cohen-Macaulay-Moduln. Math. Nachr. 85, 57–73 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Seiler, W.: A combinatorial approach to involution and δ-regularity I: Involutive bases in polynomial algebras of solvable type. Appl. Alg. Eng. Comm. Comp. 20, 207–259 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Seiler, W.: A combinatorial approach to involution and δ-regularity II: Structure analysis of polynomial modules with Pommaret bases. Appl. Alg. Eng. Comm. Comp. 20, 261–338 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Seiler, W.: Involution — The Formal Theory of Differential Equations and its Applications in Computer Algebra. Algorithms and Computation in Mathematics, vol. 24. Springer, Berlin (2009)Google Scholar
  27. 27.
    Seiler, W.: Effective genericity, δ-regularity and strong Noether position. Comm. Alg. (to appear)Google Scholar
  28. 28.
    Sharifan, L., Varbaro, M.: Graded Betti numbers of ideals with linear quotients. Matematiche 63, 257–265 (2008)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Trung, N.: Gröbner bases, local cohomology and reduction number. Proc. Amer. Math. Soc. 129, 9–18 (2001)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Amir Hashemi
    • 1
  • Michael Schweinfurter
    • Werner M. Seiler
      1. 1.Department of Mathematical SciencesIsfahan University of TechnologyIsfahanIran

      Personalised recommendations