Let \(S = K[x_1 ,\dots, x_n]\) denote the polynomial ring in \(n\) variables over a field \(K\) and let \(I\subset S\) be a homogeneous ideal. The depth function \(f_I:\mathbb{Z}_{>0}\to \mathbb{Z}_{\geq 0}\), defined by \(f_I(n) = \text{depth}(S/I^n)\), is a tool to study the behavior of the powers \(I^n\). It is a well-known result of M. Brodmann that \(f_I\) is eventually constant [Math. Proc. Camb. Philos. Soc. 86, 35–39 (1979; Zbl 0413.13011)], and J. Herzog and T. Hibi [J. Algebra 291, No. 2, 534–550 (2005; Zbl 1096.13015)] conjectured that any eventually-constant non-negative integer-valued function can be realized as the depth function of some monomial ideal.
In [Math. Z. 282, No. 3-4, 819–838 (2016; Zbl 1345.13006)], the authors gave an incorrect proof that the conjecture holds for any non-increasing, non-negative integer-valued function. The paper under review identifies this gap and in doing so proves the conjecture for a large class of non-increasing functions (Theorem 2.1).
Although, as of this writing, the preprint [H. T. Ha et al., “Symbolic powers of sums of ideals”, Preprint, arXiv:1702.01766] purports to have proven the full conjecture, the paper under review is still noteworthy for its inclusion of a characterization (Theorem 3.1) of triplets of integers \(n\geq 1\), \(d\geq 0\) and \(r\geq 1\) such that there is a monomial ideal \(I\subset S\) for which \(\lim_{k\to \infty} f_I(k)=d\) and \(r\geq 1\) is the smallest integer such that \(f_I(r) = f_I({r+1}) = f_I({r+2})=\cdots\).
Moreover, the proof of Theorem 2.1 is constructive and draws on the theory of Buchberger graphs, and as such may be of independent interest.