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ON THE ASSOCIATED PRIME IDEALS AND THE DEPTH OF POWERS OF SQUAREFREE PRINCIPAL BOREL IDEALS

Year 2019, , 224 - 244, 11.07.2019
https://doi.org/10.24330/ieja.587081

Abstract

We study algebraic properties of powers of squarefree principal
Borel ideals I, and show that astab(I) = dstab(I). Furthermore, the behaviour
of the depth function depth S/I^k is considered.

References

  • C. Andrei, V. Ene and B. Lajmiri, Powers of t-spread principal Borel ideals, Arch. Math. (Basel), 112(6) (2019), 587-597.
  • A. Aramova, J. Herzog and T. Hibi, Squarefree lexsegment ideals, Math. Z., 228(2) (1998), 353-378.
  • A. Aslam, The stable set of associated prime ideals of a squarefree principal Borel ideal, Bull. Math. Soc. Sci. Math. Roumanie (N.S.), 57(105) (2014), 243- 252.
  • M. Brodmann, Asymptotic stability of Ass(M=InM), Proc. Amer. Math. Soc., 74(1) (1979), 16-18.
  • M. Brodmann, The asymptotic nature of the analytic spread, Math. Proc. Cambridge Philos. Soc., 86 (1979), 35-39.
  • W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge Studies in Advanced Mathematics, 39, Cambridge University Press, Cambridge, 1993.
  • E. De Negri, Toric rings generated by special stable sets of monomials, Math. Nachr., 203(1) (1999), 31-45.
  • D. Eisenbud, Commutative Algebra: with a view toward algebraic geometry, Graduate Texts in Mathematics, 150, Springer-Verlag, New York, 1995.
  • V. Ene and J. Herzog, Grobner Bases in Commutative Algebra, Graduate Studies in Mathematics, 130, American Mathematical Society, Providence, RI, 2012.
  • V. Ene, J. Herzog and A. Asloob Qureshi, t-spread strongly stable ideals, arXiv:1805.02368 [math.AC].
  • C. A. Francisco, Minimal graded Betti numbers and stable ideals, Comm. Algebra, 31(10) (2003), 4971-4987.
  • C. A. Francisco, J. Mermin and J. Schweig, Borel generators, J. Algebra, 332(1) (2011), 522-542.
  • J. Herzog and T. Hibi, The depth of powers of an ideal, J. Algebra, 291(2) (2005), 534-550.
  • J. Herzog and T. Hibi, Monomial Ideals, Graduate Texts in Mathematics, 260, Springer-Verlag London, Ltd., London, 2011.
  • J. Herzog and T. Hibi, Bounding the socles of powers of squarefree monomial ideals, Commutative Algebra and Noncommutative Algebraic Geometry, Vol. II, Math. Sci. Res. Inst. Publ., 68, Cambridge Univ. Press, New York, (2015), 223-229.
  • J. Herzog, A. Rauf and M. Vladoiu, The stable set of associated prime ideals of a polymatroidal ideal, J. Algebraic Combin., 37(2) (2013), 289-312.
  • G. Kalai, Algebraic shifting, in: Computational Commutative Algebra and Combinatorics, (Osaka, 1999), Adv. Stud. Pure Math., 33, Math. Soc. Japan, Tokyo, (2002), 121-163.
  • I. Peeva and M. Stillman, The minimal free resolution of a Borel ideal, Expo. Math., 26(3) (2008), 237-247.
Year 2019, , 224 - 244, 11.07.2019
https://doi.org/10.24330/ieja.587081

Abstract

References

  • C. Andrei, V. Ene and B. Lajmiri, Powers of t-spread principal Borel ideals, Arch. Math. (Basel), 112(6) (2019), 587-597.
  • A. Aramova, J. Herzog and T. Hibi, Squarefree lexsegment ideals, Math. Z., 228(2) (1998), 353-378.
  • A. Aslam, The stable set of associated prime ideals of a squarefree principal Borel ideal, Bull. Math. Soc. Sci. Math. Roumanie (N.S.), 57(105) (2014), 243- 252.
  • M. Brodmann, Asymptotic stability of Ass(M=InM), Proc. Amer. Math. Soc., 74(1) (1979), 16-18.
  • M. Brodmann, The asymptotic nature of the analytic spread, Math. Proc. Cambridge Philos. Soc., 86 (1979), 35-39.
  • W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge Studies in Advanced Mathematics, 39, Cambridge University Press, Cambridge, 1993.
  • E. De Negri, Toric rings generated by special stable sets of monomials, Math. Nachr., 203(1) (1999), 31-45.
  • D. Eisenbud, Commutative Algebra: with a view toward algebraic geometry, Graduate Texts in Mathematics, 150, Springer-Verlag, New York, 1995.
  • V. Ene and J. Herzog, Grobner Bases in Commutative Algebra, Graduate Studies in Mathematics, 130, American Mathematical Society, Providence, RI, 2012.
  • V. Ene, J. Herzog and A. Asloob Qureshi, t-spread strongly stable ideals, arXiv:1805.02368 [math.AC].
  • C. A. Francisco, Minimal graded Betti numbers and stable ideals, Comm. Algebra, 31(10) (2003), 4971-4987.
  • C. A. Francisco, J. Mermin and J. Schweig, Borel generators, J. Algebra, 332(1) (2011), 522-542.
  • J. Herzog and T. Hibi, The depth of powers of an ideal, J. Algebra, 291(2) (2005), 534-550.
  • J. Herzog and T. Hibi, Monomial Ideals, Graduate Texts in Mathematics, 260, Springer-Verlag London, Ltd., London, 2011.
  • J. Herzog and T. Hibi, Bounding the socles of powers of squarefree monomial ideals, Commutative Algebra and Noncommutative Algebraic Geometry, Vol. II, Math. Sci. Res. Inst. Publ., 68, Cambridge Univ. Press, New York, (2015), 223-229.
  • J. Herzog, A. Rauf and M. Vladoiu, The stable set of associated prime ideals of a polymatroidal ideal, J. Algebraic Combin., 37(2) (2013), 289-312.
  • G. Kalai, Algebraic shifting, in: Computational Commutative Algebra and Combinatorics, (Osaka, 1999), Adv. Stud. Pure Math., 33, Math. Soc. Japan, Tokyo, (2002), 121-163.
  • I. Peeva and M. Stillman, The minimal free resolution of a Borel ideal, Expo. Math., 26(3) (2008), 237-247.
There are 18 citations in total.

Details

Primary Language English
Journal Section Articles
Authors

Jurgen Herzog This is me

Bahareh Lajmiri This is me

Farhad Rahmati This is me

Publication Date July 11, 2019
Published in Issue Year 2019

Cite

APA Herzog, J., Lajmiri, B., & Rahmati, F. (2019). ON THE ASSOCIATED PRIME IDEALS AND THE DEPTH OF POWERS OF SQUAREFREE PRINCIPAL BOREL IDEALS. International Electronic Journal of Algebra, 26(26), 224-244. https://doi.org/10.24330/ieja.587081
AMA Herzog J, Lajmiri B, Rahmati F. ON THE ASSOCIATED PRIME IDEALS AND THE DEPTH OF POWERS OF SQUAREFREE PRINCIPAL BOREL IDEALS. IEJA. July 2019;26(26):224-244. doi:10.24330/ieja.587081
Chicago Herzog, Jurgen, Bahareh Lajmiri, and Farhad Rahmati. “ON THE ASSOCIATED PRIME IDEALS AND THE DEPTH OF POWERS OF SQUAREFREE PRINCIPAL BOREL IDEALS”. International Electronic Journal of Algebra 26, no. 26 (July 2019): 224-44. https://doi.org/10.24330/ieja.587081.
EndNote Herzog J, Lajmiri B, Rahmati F (July 1, 2019) ON THE ASSOCIATED PRIME IDEALS AND THE DEPTH OF POWERS OF SQUAREFREE PRINCIPAL BOREL IDEALS. International Electronic Journal of Algebra 26 26 224–244.
IEEE J. Herzog, B. Lajmiri, and F. Rahmati, “ON THE ASSOCIATED PRIME IDEALS AND THE DEPTH OF POWERS OF SQUAREFREE PRINCIPAL BOREL IDEALS”, IEJA, vol. 26, no. 26, pp. 224–244, 2019, doi: 10.24330/ieja.587081.
ISNAD Herzog, Jurgen et al. “ON THE ASSOCIATED PRIME IDEALS AND THE DEPTH OF POWERS OF SQUAREFREE PRINCIPAL BOREL IDEALS”. International Electronic Journal of Algebra 26/26 (July 2019), 224-244. https://doi.org/10.24330/ieja.587081.
JAMA Herzog J, Lajmiri B, Rahmati F. ON THE ASSOCIATED PRIME IDEALS AND THE DEPTH OF POWERS OF SQUAREFREE PRINCIPAL BOREL IDEALS. IEJA. 2019;26:224–244.
MLA Herzog, Jurgen et al. “ON THE ASSOCIATED PRIME IDEALS AND THE DEPTH OF POWERS OF SQUAREFREE PRINCIPAL BOREL IDEALS”. International Electronic Journal of Algebra, vol. 26, no. 26, 2019, pp. 224-4, doi:10.24330/ieja.587081.
Vancouver Herzog J, Lajmiri B, Rahmati F. ON THE ASSOCIATED PRIME IDEALS AND THE DEPTH OF POWERS OF SQUAREFREE PRINCIPAL BOREL IDEALS. IEJA. 2019;26(26):224-4.