Depth functions of powers of homogeneous ideals

Abstract

We settle a conjecture of Herzog and Hibi, which states that the function d e p t h S / Q n \mathrm {depth}\, S/Q^n , n ≥ 1 n \ge 1 , where Q Q is a homogeneous ideal in a polynomial ring S 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.