## Correction to: Invent. math. https://doi.org/10.1007/s00222-019-00897-y

The original proof of Theorem 3.3 is incorrectly claims that \(\overline{I^{(t)}} = \bigcap _{F \in {\mathcal {F}}(I)} \overline{I_F^t}\). We have found a counter-example to this claim. The proof remains correct if we replace \(\overline{I^{(t)}}\) by \(\bigcap _{F \in {\mathcal {F}}(I)} \overline{I_F^t}\) for all \(t \ge 1\); see the corrected proof below. The correction concerns only this proof and does not affect any result of the paper.

### Theorem 3.3

Let *I* be a monomial ideal in *R* such that \(I^{(t)}\) is integrally closed for \(t \gg 0\). Then \({{\,\mathrm{depth}\,}}R/I^{(t)}\) is a convergent function with

which is also the minimum of \({{\,\mathrm{depth}\,}}R/I^{(t)}\) among all integrally closed symbolic powers \(I^{(t)}\).

### Proof

Let *m* be the minimum of \({{\,\mathrm{depth}\,}}R/I^{(t)}\) among all integrally closed symbolic powers \(I^{(t)}\). Choose an integrally closed symbolic power \(I^{(s)}\) such that \({{\,\mathrm{depth}\,}}R/I^{(s)} = m\). By Theorem 2.6(ii), there exists an integer *a* such that \({{\,\mathrm{depth}\,}}R/I^{(s)}\ge {{\,\mathrm{depth}\,}}R/I^{(t)}\) for \(t \ge as^2\). This implies \({{\,\mathrm{depth}\,}}R/I^{(t)} = m\) for all integrally closed symbolic powers \(I^{(t)}\) with \(t \ge as^2\). Since \(I^{(t)}\) is integrally closed for \(t \gg 0\), we get \({{\,\mathrm{depth}\,}}R/I^{(t)} = m\) for \(t \gg 0\).

Let \(I_t = \bigcap _{F \in {\mathcal {F}}(I)} \overline{I_F^t}.\) We will show that \(m = \min _{t \ge 1} {{\,\mathrm{depth}\,}}R/I_t\). Since \(I_t = I^{(t)}\) for \(t \gg 0\) by Proposition 2.2,

By Proposition 2.3, we have

for all \(s, t \ge 1\). 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

Therefore,

for all \(s \ge 1\). Now, we can conclude that

It remains to show that \(\min _{t \ge 1} {{\,\mathrm{depth}\,}}R/I_t = \dim R - \dim F_s(I).\) For that, we need the following auxiliary observation (cf. [41, Proposition 2.5]).

Let \({\mathcal {F}}\) denote the filtration of the ideals \(I_t\), \(t \ge 0\). Let \(R({\mathcal {F}}) = \bigoplus _{t \ge 0}I_ty^t\). Then \(R({\mathcal {F}})\) is an algebra generated by monomials in \(k[x_1,...,x_n,y]\). We have

For each \(F \in {\mathcal {F}}(I)\), the algebra \(\bigoplus _{t \ge 0}\overline{I_F^{t}}y^t\) is the normalization of the finitely generated algebra \(\bigoplus _{t \ge 0}I_F^{t}y^t\). Hence, \(\bigoplus _{t \ge 0}\overline{I_F^{t}}y^t\) is a finitely generated algebra. The monomials of \(\bigoplus _{t \ge 0}\overline{I_F^{t}}y^t\) 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 \(G({\mathcal {F}}) = \bigoplus _{t \ge 0}I_t/I_{t+1}\). Then \(G({\mathcal {F}})\) is a factor ring of \(R({\mathcal {F}})\) by the ideal \(\bigoplus _{t \ge 0}I_{t+1}y^t\). 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

We have \(G({\mathcal {F}})/{\mathfrak {m}}G({\mathcal {F}}) = \bigoplus _{t \ge 0}I_t/({\mathfrak {m}}I_t+I_{t+1})\). Since \(F_s(I) = \bigoplus _{t \ge 0} I^{(t)}/{\mathfrak {m}}I^{(t)}\), \(I^{(t+1)} \subseteq {\mathfrak {m}}I^{(t)}\) [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,

\(\square \)

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

## Acknowledgements

The original paper and this correction 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.

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Nguyen, H.D., Trung, N.V. Correction to: Depth functions of symbolic powers of homogeneous ideals.
*Invent. math.* **218, **829–831 (2019). https://doi.org/10.1007/s00222-019-00925-x

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DOI: https://doi.org/10.1007/s00222-019-00925-x