The Stacks project

Lemma 41.21.7. Let $X$ be a locally Noetherian scheme. Let $D \subset X$ be a closed subscheme. If $X$ is J-2 or Nagata, then following are equivalent

  1. $D$ is a normal crossings divisor in $X$,

  2. for every $p \in D$ the pullback of $D$ to the spectrum of the strict henselization $\mathcal{O}_{X, p}^{sh}$ is a strict normal crossings divisor.

Proof. The implication (1) $\Rightarrow $ (2) is straightforward and does not need the assumption that $X$ is J-2 or Nagata. Namely, let $p \in D$ and choose an étale neighbourhood $(U, u) \to (X, p)$ such that the pullback of $D$ is a strict normal crossings divisor on $U$. Then $\mathcal{O}_{X, p}^{sh} = \mathcal{O}_{U, u}^{sh}$ and we see that the trace of $D$ on $\mathop{\mathrm{Spec}}(\mathcal{O}_{U, u}^{sh})$ is cut out by part of a regular system of parameters as this is already the case in $\mathcal{O}_{U, u}$.

To prove the implication in the other direction we will use the criterion of Lemma 41.21.6. Observe that formation of the normalization $D^\nu \to D$ commutes with strict henselization, see More on Morphisms, Lemma 37.19.4. If we can show that $D^\nu \to D$ is finite, then we see that $D^\nu \to D$ and the schemes $Z_ n$ satisfy all desired properties because these can all be checked on the level of local rings (but the finiteness of the morphism $D^\nu \to D$ is not something we can check on local rings). We omit the detailed verifications.

If $X$ is Nagata, then $D^\nu \to D$ is finite by Morphisms, Lemma 29.54.10.

Assume $X$ is J-2. Choose a point $p \in D$. We will show that $D^\nu \to D$ is finite over a neighbourhood of $p$. By assumption there exists a regular system of parameters $f_1, \ldots , f_ d$ of $\mathcal{O}_{X, p}^{sh}$ and $1 \leq r \leq d$ such that the trace of $D$ on $\mathop{\mathrm{Spec}}(\mathcal{O}_{X, p}^{sh})$ is cut out by $f_1 \ldots f_ r$. Then

\[ D^\nu \times _ X \mathop{\mathrm{Spec}}(\mathcal{O}_{X, p}^{sh}) = \coprod \nolimits _{i = 1, \ldots , r} V(f_ i) \]

Choose an affine étale neighbourhood $(U, u) \to (X, p)$ such that $f_ i$ comes from $f_ i \in \mathcal{O}_ U(U)$. Set $D_ i = V(f_ i) \subset U$. The strict henselization of $\mathcal{O}_{D_ i, u}$ is $\mathcal{O}_{X, p}^{sh}/(f_ i)$ which is regular. Hence $\mathcal{O}_{D_ i, u}$ is regular (for example by More on Algebra, Lemma 15.45.10). Because $X$ is J-2 the regular locus is open in $D_ i$. Thus after replacing $U$ by a Zariski open we may assume that $D_ i$ is regular for each $i$. It follows that

\[ \coprod \nolimits _{i = 1, \ldots , r} D_ i = D^\nu \times _ X U \longrightarrow D \times _ X U \]

is the normalization morphism and it is clearly finite. In other words, we have found an étale neighbourhood $(U, u)$ of $(X, p)$ such that the base change of $D^\nu \to D$ to this neighbourhood is finite. This implies $D^\nu \to D$ is finite by descent (Descent, Lemma 35.23.23) and the proof is complete. $\square$

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