Depth functions of symbolic powers of homogeneous ideals (1907.06468v2)

Abstract: This paper addresses the problem of comparing minimal free resolutions of symbolic powers of an ideal. Our investigation is focused on the behavior of the function depth R/I^t = dim R - pd I^t - 1, where I^t denotes the t-th symbolic power of a homogeneous ideal I in a noetherian polynomial ring R and pd denotes the projective dimension. It has been an open question whether the function depth R/I^t is non-increasing if I is a squarefree monomial ideal. We show that depth R/I^t is almost non-increasing in the sense that depth R/I^s \ge depth R/I^t for all s \ge 1 and t \in E(s), where E(s) = \cup_{i \ge 1} {t \in N| i(s-1)+1 \le t \le is} (which contains all integers t \ge (s-1)^2+1). The range E(s) is the best possible since we can find squarefree monomial ideals I such that depth R/I^s < depth R/I^t for t \not\in E(s), which gives a negative answer to the above question. Another open question asks whether the function depth R/I^t is always constant for t \gg 0. We are able to construct counter-examples to this question by monomial ideals. On the other hand, we show that if I is a monomial ideal such that I^t is integrally closed for t \gg 0 (e.g. if I is a squarefree monomial ideal), then depth R/I^t is constant for t \gg 0 with lim_{t \to \infty} depth R/I^t = dim R - dim \oplus_{t \ge 0} I^t/m I^t. Our last result (which is the main contribution of this paper) shows that for any positive numerical function \phi(t) which is periodic for t \gg 0, there exist a polynomial ring R and a homogeneous ideal I such that depth R/I^t = \phi(t) for all t \ge 1. As a consequence, for any non-negative numerical function \psi(t) which is periodic for t \gg 0, there is a homogeneous ideal I and a number c such that pd I^t = \psi(t) + c for all t \ge 1.