Depth functions of powers of homogeneous ideals

Hà, Huy; Nguyen, Hop; Trung, Ngo; Trung, Tran

doi:10.1090/proc/15083

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.

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