The Stacks project

Lemma 41.21.6. Let $X$ be a locally Noetherian scheme. Let $D \subset X$ be a closed subscheme. The following are equivalent

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

  2. $D$ is reduced, the normalization $\nu : D^\nu \to D$ is unramified, and for any $n \geq 1$ the scheme

    \[ Z_ n = D^\nu \times _ D \ldots \times _ D D^\nu \setminus \{ (p_1, \ldots , p_ n) \mid p_ i = p_ j\text{ for some }i\not= j\} \]

    is regular, the morphism $Z_ n \to X$ is a local complete intersection morphism whose conormal sheaf is locally free of rank $n$.

Proof. First we explain how to think about condition (2). The diagonal of an unramified morphism is open (Morphisms, Lemma 29.35.13). On the other hand $D^\nu \to D$ is separated, hence the diagonal $D^\nu \to D^\nu \times _ D D^\nu $ is closed. Thus $Z_ n$ is an open and closed subscheme of $D^\nu \times _ D \ldots \times _ D D^\nu $. On the other hand, $Z_ n \to X$ is unramified as it is the composition

\[ Z_ n \to D^\nu \times _ D \ldots \times _ D D^\nu \to \ldots \to D^\nu \times _ D D^\nu \to D^\nu \to D \to X \]

and each of the arrows is unramified. Since an unramified morphism is formally unramified (More on Morphisms, Lemma 37.6.8) we have a conormal sheaf $\mathcal{C}_ n = \mathcal{C}_{Z_ n/X}$ of $Z_ n \to X$, see More on Morphisms, Definition 37.7.2.

Formation of normalization commutes with étale localization by More on Morphisms, Lemma 37.19.3. Checking that local rings are regular, or that a morphism is unramified, or that a morphism is a local complete intersection or that a morphism is unramified and has a conormal sheaf which is locally free of a given rank, may be done étale locally (see More on Algebra, Lemma 15.44.3, Descent, Lemma 35.23.28, More on Morphisms, Lemma 37.62.19 and Descent, Lemma 35.7.6).

By the remark of the preceding paragraph and the definition of normal crossings divisor it suffices to prove that a strict normal crossings divisor $D = \bigcup _{i \in I} D_ i$ satisfies (2). In this case $D^\nu = \coprod D_ i$ and $D^\nu \to D$ is unramified (being unramified is local on the source and $D_ i \to D$ is a closed immersion which is unramified). Similarly, $Z_1 = D^\nu \to X$ is a local complete intersection morphism because we may check this locally on the source and each morphism $D_ i \to X$ is a regular immersion as it is the inclusion of a Cartier divisor (see Lemma 41.21.2 and More on Morphisms, Lemma 37.62.9). Since an effective Cartier divisor has an invertible conormal sheaf, we conclude that the requirement on the conormal sheaf is satisfied. Similarly, the scheme $Z_ n$ for $n \geq 2$ is the disjoint union of the schemes $D_ J = \bigcap _{j \in J} D_ j$ where $J \subset I$ runs over the subsets of order $n$. Since $D_ J \to X$ is a regular immersion of codimension $n$ (by the definition of strict normal crossings and the fact that we may check this on stalks by Divisors, Lemma 31.20.8) it follows in the same manner that $Z_ n \to X$ has the required properties. Some details omitted.

Assume (2). Let $p \in D$. Since $D^\nu \to D$ is unramified, it is finite (by Morphisms, Lemma 29.44.4). Hence $D^\nu \to X$ is finite unramified. By Lemma 41.17.3 and étale localization (permissible by the discussion in the second paragraph and the definition of normal crossings divisors) we reduce to the case where $D^\nu = \coprod _{i \in I} D_ i$ with $I$ finite and $D_ i \to U$ a closed immersion. After shrinking $X$ if necessary, we may assume $p \in D_ i$ for all $i \in I$. The condition that $Z_1 = D^\nu \to X$ is an unramified local complete intersection morphism with conormal sheaf locally free of rank $1$ implies that $D_ i \subset X$ is an effective Cartier divisor, see More on Morphisms, Lemma 37.62.3 and Divisors, Lemma 31.21.3. To finish the proof we may assume $X = \mathop{\mathrm{Spec}}(A)$ is affine and $D_ i = V(f_ i)$ with $f_ i \in A$ a nonzerodivisor. If $I = \{ 1, \ldots , r\} $, then $p \in Z_ r = V(f_1, \ldots , f_ r)$. The same reference as above implies that $(f_1, \ldots , f_ r)$ is a Koszul regular ideal in $A$. Since the conormal sheaf has rank $r$, we see that $f_1, \ldots , f_ r$ is a minimal set of generators of the ideal defining $Z_ r$ in $\mathcal{O}_{X, p}$. This implies that $f_1, \ldots , f_ r$ is a regular sequence in $\mathcal{O}_{X, p}$ such that $\mathcal{O}_{X, p}/(f_1, \ldots , f_ r)$ is regular. Thus we conclude by Algebra, Lemma 10.106.7 that $f_1, \ldots , f_ r$ can be extended to a regular system of parameters in $\mathcal{O}_{X, p}$ and this finishes the proof. $\square$

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