Depth functions of symbolic powers of homogeneous ideals

TL;DR

This paper investigates the behavior of depth functions of symbolic powers of homogeneous ideals, providing counterexamples to conjectures about their monotonicity and constancy, and constructing ideals with prescribed periodic depth functions.

Contribution

It demonstrates that depth functions can be non-monotonic and non-constant, and constructs ideals with any periodic depth behavior, advancing understanding of symbolic power resolutions.

Findings

Depth R/I^(t) is almost non-increasing for squarefree monomial ideals.

Counterexamples show depth R/I^(t) is not always constant for large t.

Existence of ideals with prescribed periodic depth functions.

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.

Equations

$$\underset{t\to \infty }{\lim }\mathrm{depth}R/I^{(t)}=\dim R-\dim F_s(I),$$

$$m\ge \underset{t\ge 1}{\min }\mathrm{depth}R/I_t.$$

$$\operatorname{depth}R/I_{s}\geq\operatorname{depth}R/\big{(}I_{s})^{(t)}$$

$$(I_s{)}^{(t)}=\bigcap _{F\in \mathcal{F}(I)}{I}_F^{st}=I^{(st)}.$$

$$\mathrm{depth}R/I_s\ge \mathrm{depth}R/I^{(st)}\ge m$$

$$m=\underset{t\ge 1}{\min }\mathrm{depth}R/I_t.$$

$$R(\mathcal{F})=\bigcap _{F\in \mathcal{F}(I)}\underset{t\ge 0}{⨁}\overline{I_F^t}{y}^t.$$

$$\underset{t\ge 1}{\min }\mathrm{depth}R/I_t$$

$$=\dim G(\mathcal{F})-\dim G(\mathcal{F})/\mathfrak{m}G(\mathcal{F})$$

$$=\dim R-\dim G(\mathcal{F})/\mathfrak{m}G(\mathcal{F}).$$

$$\underset{t\ge 1}{\min }\mathrm{depth}R/I_t=\dim R-\dim F_s(I).$$

Full text

Let $I_{t}=\bigcap_{F\in{\mathcal{F}}(I)}\overline{I_{F}^{t}}.$ We will show that $m=\min_{t\geq 1}\operatorname{depth}R/I_{t}$. Since $I_{t}=I^{(t)}$ for $t\gg 0$ by Proposition 2.2,

[TABLE]

By Proposition 2.3, we have

[TABLE]

for all $s,t\geq 1$. For $t\gg 0$, $I^{(st)}$ is integrally closed and so is $I_{F}^{st}$ for all $F\in{\mathcal{F}}(I)$ by Proposition 2.2. This implies I_{F}^{st}\subseteq\big{(}\overline{I_{F}^{s}}\big{)}^{t}\subseteq\overline{I_{F}^{st}}=I_{F}^{st}. Hence, \big{(}\overline{I_{F}^{s}}\big{)}^{t}=I_{F}^{st}. So we get

[TABLE]

Therefore,

[TABLE]

for all $s\geq 1$. Now, we can conclude that

[TABLE]

It remains to show that $\min_{t\geq 1}\operatorname{depth}R/I_{t}=\dim R-\dim F_{s}(I).$ For that we need the following auxiliary observation (cf. [41, Proposition 2.5]).

Let ${\mathcal{F}}$ denote the filtration of the ideals $I_{t}$, $t\geq 0$. Let $R({\mathcal{F}})=\bigoplus_{t\geq 0}I_{t}y^{t}$. Then $R({\mathcal{F}})$ is an algebra generated by monomials in $k[x_{1},...,x_{n},y]$. We have

[TABLE]

For each $F\in{\mathcal{F}}(I)$, the algebra $\bigoplus_{t\geq 0}\overline{I_{F}^{t}}y^{t}$ is the normalization of the finitely generated algebra $\bigoplus_{t\geq 0}I_{F}^{t}y^{t}$. Hence, $\bigoplus_{t\geq 0}\overline{I_{F}^{t}}y^{t}$ is a finitely generated algebra. The monomials of $\bigoplus_{t\geq 0}\overline{I_{F}^{t}}y^{t}$ form a finitely generated semigroup. Since the semigroup of the monomials of $R({\mathcal{F}})$ is the intersections of these semigroups, it is also finitely generated [14, Corollary 1.2]. From this it follows that $R({\mathcal{F}})$ is a finitely generated algebra. Moreover, as an intersection of normal rings, $R({\mathcal{F}})$ is a normal ring. By [20, Theorem 1], this implies that $R({\mathcal{F}})$ is Cohen-Macaulay.

Let $G({\mathcal{F}})=\bigoplus_{t\geq 0}I_{t}/I_{t+1}$. Then $G({\mathcal{F}})$ is a factor ring of $R({\mathcal{F}})$ by the ideal $\bigoplus_{t\geq 0}I_{t+1}y^{t}$. Hence, $G({\mathcal{F}})$ is a finitely generated algebra. By [4, Theorem 4.5.6(b)], we have $\dim G({\mathcal{F}})=\dim R$. By the proof of the necessary part of [40, Theorem 1.1], the Cohen-Macaulayness of $R({\mathcal{F}})$ implies that of $G({\mathcal{F}})$. By [5, Theorem 9.23], these facts imply

[TABLE]

We have $G({\mathcal{F}})/{\mathfrak{m}}G({\mathcal{F}})=\bigoplus_{t\geq 0}I_{t}/({\mathfrak{m}}I_{t}+I_{t+1})$. Since $F_{s}(I)=\bigoplus_{t\geq 0}I^{(t)}/{\mathfrak{m}}I^{(t)}$, $I^{(t+1)}\subseteq{\mathfrak{m}}I^{(t)}$ [10, Proposition 9] and $I_{t}=I^{(t)}$ for $t\gg 0$, the graded algebras $G({\mathcal{F}})/{\mathfrak{m}}G({\mathcal{F}})$ and $F_{s}(I)$ share the same Hilbert quasi-polynomial [4, Theorem 4.4.3]. From this it follows that $\dim G({\mathcal{F}})/{\mathfrak{m}}G({\mathcal{F}})=\dim F_{s}(I)$. Therefore,

[TABLE]

∎

Moreover, the reference [40] lists the wrong year. It has to be 1989 instead of 1997.

Acknowledgement. The original paper and this corrigendum are supported by Vietnam National Foundation for Science and Technology Development under grant number 101.04-2019.313. The authors thank Arvind Kumar for pointing out the mistake of the original proof.