## Depth functions of powers of homogeneous ideals

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- by Huy Tài Hà, Hop Dang Nguyen, Ngo Viet Trung and Tran Nam Trung
- Proc. Amer. Math. Soc.
**149**(2021), 1837-1844 - DOI: https://doi.org/10.1090/proc/15083
- Published electronically: March 2, 2021
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## Abstract:

We settle a conjecture of Herzog and Hibi, which states that the function $\mathrm {depth} S/Q^n$, $n \ge 1$, where $Q$ is a homogeneous ideal in a polynomial ring $S$, can be any convergent numerical function. We also give a positive answer to a longstanding open question of Ratliff on the associated primes of powers of ideals.## References

- Somayeh Bandari, Jürgen Herzog, and Takayuki Hibi,
*Monomial ideals whose depth function has any given number of strict local maxima*, Ark. Mat.**52**(2014), no. 1, 11–19. MR**3175291**, DOI 10.1007/s11512-013-0184-1 - M. Brodmann,
*The asymptotic nature of the analytic spread*, Math. Proc. Cambridge Philos. Soc.**86**(1979), no. 1, 35–39. MR**530808**, DOI 10.1017/S030500410000061X - Winfried Bruns and Jürgen Herzog,
*Cohen-Macaulay rings*, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR**1251956** - David Eisenbud,
*Commutative algebra*, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR**1322960**, DOI 10.1007/978-1-4612-5350-1 - Huy Tài Hà, Hop Dang Nguyen, Ngo Viet Trung, and Tran Nam Trung,
*Symbolic powers of sums of ideals*, Math. Z.**294**(2020), no. 3-4, 1499–1520. MR**4074049**, DOI 10.1007/s00209-019-02323-8 - Huy Tài Hà and Mengyao Sun,
*Squarefree monomial ideals that fail the persistence property and non-increasing depth*, Acta Math. Vietnam.**40**(2015), no. 1, 125–137. MR**3331937**, DOI 10.1007/s40306-014-0104-x - Huy Tài Hà, Ngo Viet Trung, and Trân Nam Trung,
*Depth and regularity of powers of sums of ideals*, Math. Z.**282**(2016), no. 3-4, 819–838. MR**3473645**, DOI 10.1007/s00209-015-1566-9 - J. Herzog,
*Algebraic and homological properties of powers and symbolic powers of ideals*, Lect. Notes, CIMPA School on Combinatorial and Computational Aspects of Commutative Algebra, Lahore, 2009, http://www.cimpa-icpam.org/IMG/pdf/LahoreHerzog.pdf. - Jürgen Herzog and Takayuki Hibi,
*The depth of powers of an ideal*, J. Algebra**291**(2005), no. 2, 534–550. MR**2163482**, DOI 10.1016/j.jalgebra.2005.04.007 - Le Tuan Hoa and Nguyen Duc Tam,
*On some invariants of a mixed product of ideals*, Arch. Math. (Basel)**94**(2010), no. 4, 327–337. MR**2643966**, DOI 10.1007/s00013-010-0112-6 - Sam Huckaba,
*On linear equivalence of the $P$-adic and $P$-symbolic topologies*, J. Pure Appl. Algebra**46**(1987), no. 2-3, 179–185. MR**897014**, DOI 10.1016/0022-4049(87)90092-2 - Kazunori Matsuda, Tao Suzuki, and Akiyoshi Tsuchiya,
*Nonincreasing depth functions of monomial ideals*, Glasg. Math. J.**60**(2018), no. 2, 505–511. MR**3784062**, DOI 10.1017/S0017089517000349 - Susan Morey and Rafael H. Villarreal,
*Edge ideals: algebraic and combinatorial properties*, Progress in commutative algebra 1, de Gruyter, Berlin, 2012, pp. 85–126. MR**2932582** - Hop Dang Nguyen and Ngo Viet Trung,
*Depth functions of symbolic powers of homogeneous ideals*, Invent. Math.**218**(2019), no. 3, 779–827. MR**4022079**, DOI 10.1007/s00222-019-00897-y - L. J. Ratliff Jr.,
*A brief survey and history of asymptotic prime divisors*, Rocky Mountain J. Math.**13**(1983), no. 3, 437–459. MR**715767**, DOI 10.1216/RMJ-1983-13-3-437

## Bibliographic Information

**Huy Tài Hà**- Affiliation: Department of Mathematics, Tulane University. 6823 St. Charles Avenue, New Orleans, Louisiana 70118
- ORCID: 0000-0002-6002-3453
- Email: tha@tulane.edu
**Hop Dang Nguyen**- Affiliation: Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Hanoi, Vietnam
- MR Author ID: 981901
- Email: ngdhop@gmail.com
**Ngo Viet Trung**- Affiliation: International Centre for Research and Postgraduate Training, Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Hanoi, Vietnam
- MR Author ID: 207806
- Email: nvtrung@math.ac.vn
**Tran Nam Trung**- Affiliation: Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Hanoi, Vietnam
- Email: tntrung@math.ac.vn
- Received by editor(s): April 17, 2019
- Received by editor(s) in revised form: February 24, 2020
- Published electronically: March 2, 2021
- Additional Notes: The first author was partially supported by the Simons Foundation (grant #279786) and Louisiana Board of Regents (grant #LEQSF(2017-19)-ENH-TR-25).

The third author was partially supported by Vietnam National Foundation for Science and Technology Development under grant number 101.04-2019.313.

The second and fourth authors were partially funded by International Centre for Research and Postgraduate Training in Mathematics (ICRTM) under grant numbers ICRTM01_2020.05 and ICRTM01_2020.04. - Communicated by: Claudia Polini
- © Copyright 2021 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**149**(2021), 1837-1844 - MSC (2020): Primary 13C15, 13D02, 14B05
- DOI: https://doi.org/10.1090/proc/15083
- MathSciNet review: 4232180