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- W3048777501 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 $${{,mathrm{depth},}}R/I^{(t)} = dim R -{{,mathrm{pd},}}I^{(t)} - 1$$ , 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 $$t ge 1$$ ." @default.
- W3048777501 created "2020-08-18" @default.
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- W3048777501 date "2019-07-13" @default.
- W3048777501 modified "2023-10-05" @default.
- W3048777501 title "Depth functions of symbolic powers of homogeneous ideals" @default.
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- W3048777501 doi "https://doi.org/10.1007/s00222-019-00897-y" @default.
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