Abstract
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.
Similar content being viewed by others
References
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 (2012)
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. (2018). https://doi.org/10.1007/s10801-018-0811-9
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)
Acknowledgements
We would like to thank deeply grateful to the referee for the careful reading of the manuscript and the helpful suggestions.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Karimi, S., Mafi, A. On stability properties of powers of polymatroidal ideals. Collect. Math. 70, 357–365 (2019). https://doi.org/10.1007/s13348-018-0234-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13348-018-0234-x