LAUSR

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 \({{\,\mathrm{depth}\,}}R/I^{(t)} = \dim R -{{\,\mathrm{pd}\,}}I^{(t)} - 1\), where \(I^{(t)}\) denotes the t-th symbolic power of a homogeneous ideal I in a noetherian polynomial ring R and \({{\,\mathrm{pd}\,}}\) denotes the projective dimension. It has been an open question whether the function \({{\,\mathrm{depth}\,}}R/I^{(t)}\) is non-increasing if I is a squarefree monomial ideal. We show that \({{\,\mathrm{depth}\,}}R/I^{(t)}\) is almost non-increasing in the sense that \({{\,\mathrm{depth}\,}}R/I^{(s)} \ge {{\,\mathrm{depth}\,}}R/I^{(t)}\) for all \(s \ge 1\) and \(t \in E(s)\), where $$\begin{aligned} E(s) = \bigcup _{i \ge 1}\{t \in {\mathbb {N}}|\ i(s-1)+1 \le t \le is\} \end{aligned}$$ (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 \({{\,\mathrm{depth}\,}}R/I^{(s)} < {{\,\mathrm{depth}\,}}R/I^{(t)}\) for \(t \not \in E(s)\), which gives a negative answer to the above question. Another open question asks whether the function \({{\,\mathrm{depth}\,}}R/I^{(t)}\) 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 \({{\,\mathrm{depth}\,}}R/I^{(t)}\) is constant for \(t \gg 0\) with $$\begin{aligned} \lim _{t \rightarrow \infty }{{\,\mathrm{depth}\,}}R/I^{(t)} = \dim R - \dim \oplus _{t \ge 0}I^{(t)}/{\mathfrak {m}}I^{(t)}. \end{aligned}$$ 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 \({{\,\mathrm{depth}\,}}R/I^{(t)} = \phi (t)\) 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 \({{\,\mathrm{pd}\,}}I^{(t)} = \psi (t) + c\) for all \(t \ge 1\).