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Depth functions of powers of homogeneous ideals


Authors: Huy Tài Hà, Hop Dang Nguyen, Ngo Viet Trung and Tran Nam Trung
Journal: Proc. Amer. Math. Soc. 149 (2021), 1837-1844
MSC (2020): Primary 13C15, 13D02, 14B05
DOI: https://doi.org/10.1090/proc/15083
Published electronically: March 2, 2021
MathSciNet review: 4232180
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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.


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Additional 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

Keywords: Depth, projective dimension, associated prime, monomial ideals
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
Article copyright: © Copyright 2021 American Mathematical Society