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. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0CBS. Beware of the difference between the letter 'O' and the digit '0'.