On stability properties of powers of polymatroidal ideals

• Autores: Shokoufe Karimi, Amir Mafi
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 70, Fasc. 3, 2019, págs. 357-365
• Idioma: inglés
• DOI: 10.1007/s13348-018-0234-x
• Texto completo no disponible (Saber más ...)
• Resumen

  • Let R=K[x_1,\ldots ,x_n] be the polynomial ring in n variables over a field K with the maximal ideal \mathfrak {m}=(x_1,\ldots ,x_n). Let {\text {astab}}(I) and {\text {dstab}}(I) be the smallest integer n for which {\text {Ass}}(I^n) and {\text {depth}}(I^n) stabilize, respectively. In this paper we show that {\text {astab}}(I)={\text {dstab}}(I) in the following cases:

    (i) I is a matroidal ideal and n\le 5.

    (ii) I is a polymatroidal ideal, n=4 and \mathfrak {m}\notin {\text {Ass}}^{\infty }(I), where {\text {Ass}}^{\infty }(I) is the stable set of associated prime ideals of I.

    (iii) I is a polymatroidal ideal of degree 2.

    Moreover, we give an example of a polymatroidal ideal for which {\text {astab}}(I)\ne {\text {dstab}}(I). This is a counterexample to the conjecture of Herzog and Qureshi, according to which these two numbers are the same for polymatroidal ideals.

• Referencias bibliográficas
  • Abbott, J., Bigatti, A. M.: A C++ Library for Doing Computations in Commutative Algebra. http://cocoa.dima.unige.it/cocoalib (2018)
  • Bandari, S., Herzog, J.: Monomial localizations and polymatroidal ideals. Eur. J. Comb. 34, 752–763 (2013)
  • Bayati, S., Herzog, J., Rinaldo, G.: A routine to compute the stable set of associated prime ideals of a monomial ideal. http://www.giancarlorinaldo.it/stableset.html...
  • Brodmann, M.: Asymptotic stability of Ass (M/{I^{n}M}). Proc. Am. Math. Soc. 74, 16–18 (1979)
  • Brodmann, M.: The asymptotic nature of the analytic spread. Math. Proc. Camb. Philos. Soc. 86, 35–39 (1979)
  • Bruns, W., Herzog, J.: Cohen–Macaulay Rings. Cambridge University Press, Cambridge (1998)
  • Caviglia, G., Ha, H.T., Herzog, J., Kummini, M., Terai, N., Trung, N.V.: Depth and regularity modulo and principal ideal. J. Algebr. Comb....
  • Chiang-Hsieh, H.J.: Some arithmetic properties of matroidal ideals. Commun. Algebra 38, 944–952 (2010)
  • Conca, A., Herzog, J.: Castelnuovo–Mumford regularity of products of ideals. Collect. Math. 54, 137–152 (2003)
  • Eisenbud, D., Huneke, C.: Cohen–Macaulay Rees algebra and their specialization. J. Algebra 81, 202–224 (1983)
  • Grayson, D.R., Stillman, M.E.: Macaulay 2, a software system for research in algebraic geometry. http://www.math.uiuc.edu/Macaulay2/ (2009)
  • Herzog, J., Hibi, T.: Bounding the socles of powers of squarefree monomial ideals. Commut. Algebra Noncommut. Algebr. Geom. 68, 223–229 (2015)
  • Herzog, J., Hibi, T.: Monomial Ideals. Graduate Texts in Mathematics, vol. 260. Springer, London (2011)
  • Herzog, J., Hibi, T.: Cohen–Macaulay polymatroidal ideals. Eur. J. Comb. 27, 513–517 (2006)
  • Herzog, J., Hibi, T.: The depth of powers of an ideal. J. Algebra 291, 534–550 (2005)
  • Herzog, J., Hibi, T.: Discrete polymatroids. J. Algebr. Comb. 16, 239–268 (2002)
  • Herzog, J., Hibi, T., Vladoiu, M.: Ideals of fiber type and polymatroids. Osaka J. Math. 42, 807–829 (2005)
  • Herzog, J., Mafi, A.: Stability properties of powers of ideals in regular local rings of small dimension. Pac. J. Math. 295, 31–41 (2018)
  • Herzog, J., Qureshi, A.: Persistence and stability properties of powers of ideals. J. Pure Appl. Algebra 219, 530–542 (2015)
  • Herzog, J., Rauf, A., Vladoiu, M.: The stable set of associated prime ideals of a polymatroidal ideal. J. Algebr. Comb. 37, 289–312 (2013)
  • Herzog, J., Takayama, Y.: Resolutions by mapping cones. Homol. Homotopy Appl. 4, 277–294 (2002)
  • Herzog, J., Vladoiu, M.: Squarefree monomial ideals with constant depth function. J. Pure Appl. Algebra 217, 1764–1772 (2013)
  • Huneke, C., Swanson, I.: Integral Closure of Ideals, Rings, and Modules, vol. 13. Cambridge University Press, Cambridge (2006)
  • Katz, D., Ratliff Jr., L.R.: On the prime divisors of IJ when I is integrally closed. Arch. Math. 51, 520–522 (1988)
  • Trung, T.N.: Stability of associated primes of integral closures of monomial ideals. J. Comb. Theory Ser. A 116, 44–54 (2009)
  • Villarreal, R.H.: Monomial Algebras. Marcel Dekker Inc., New York (2001)