Definition 41.21.1. Let $X$ be a locally Noetherian scheme. A strict normal crossings divisor on $X$ is an effective Cartier divisor $D \subset X$ such that for every $p \in D$ the local ring $\mathcal{O}_{X, p}$ is regular and there exists a regular system of parameters $x_1, \ldots , x_ d \in \mathfrak m_ p$ and $1 \leq r \leq d$ such that $D$ is cut out by $x_1 \ldots x_ r$ in $\mathcal{O}_{X, p}$.
41.21 Normal crossings divisors
Here is the definition.
We often encounter effective Cartier divisors $E$ on locally Noetherian schemes $X$ such that there exists a strict normal crossings divisor $D$ with $E \subset D$ set theoretically. In this case we have $E = \sum a_ i D_ i$ with $a_ i \geq 0$ where $D = \bigcup _{i \in I} D_ i$ is the decomposition of $D$ into its irreducible components. Observe that $D' = \bigcup _{a_ i > 0} D_ i$ is a strict normal crossings divisor with $E = D'$ set theoretically. When the above happens we will say that $E$ is supported on a strict normal crossings divisor.
Lemma 41.21.2. Let $X$ be a locally Noetherian scheme. Let $D \subset X$ be an effective Cartier divisor. Let $D_ i \subset D$, $i \in I$ be its irreducible components viewed as reduced closed subschemes of $X$. The following are equivalent
$D$ is a strict normal crossings divisor, and
$D$ is reduced, each $D_ i$ is an effective Cartier divisor, and for $J \subset I$ finite the scheme theoretic intersection $D_ J = \bigcap _{j \in J} D_ j$ is a regular scheme each of whose irreducible components has codimension $|J|$ in $X$.
Proof. Assume $D$ is a strict normal crossings divisor. Pick $p \in D$ and choose a regular system of parameters $x_1, \ldots , x_ d \in \mathfrak m_ p$ and $1 \leq r \leq d$ as in Definition 41.21.1. Since $\mathcal{O}_{X, p}/(x_ i)$ is a regular local ring (and in particular a domain) we see that the irreducible components $D_1, \ldots , D_ r$ of $D$ passing through $p$ correspond $1$-to-$1$ to the height one primes $(x_1), \ldots , (x_ r)$ of $\mathcal{O}_{X, p}$. By Algebra, Lemma 10.106.3 we find that the intersections $D_{i_1} \cap \ldots \cap D_{i_ s}$ have codimension $s$ in an open neighbourhood of $p$ and that this intersection has a regular local ring at $p$. Since this holds for all $p \in D$ we conclude that (2) holds.
Assume (2). Let $p \in D$. Since $\mathcal{O}_{X, p}$ is finite dimensional we see that $p$ can be contained in at most $\dim (\mathcal{O}_{X, p})$ of the components $D_ i$. Say $p \in D_1, \ldots , D_ r$ for some $r \geq 1$. Let $x_1, \ldots , x_ r \in \mathfrak m_ p$ be local equations for $D_1, \ldots , D_ r$. Then $x_1$ is a nonzerodivisor in $\mathcal{O}_{X, p}$ and $\mathcal{O}_{X, p}/(x_1) = \mathcal{O}_{D_1, p}$ is regular. Hence $\mathcal{O}_{X, p}$ is regular, see Algebra, Lemma 10.106.7. Since $D_1 \cap \ldots \cap D_ r$ is a regular (hence normal) scheme it is a disjoint union of its irreducible components (Properties, Lemma 28.7.6). Let $Z \subset D_1 \cap \ldots \cap D_ r$ be the irreducible component containing $p$. Then $\mathcal{O}_{Z, p} = \mathcal{O}_{X, p}/(x_1, \ldots , x_ r)$ is regular of codimension $r$ (note that since we already know that $\mathcal{O}_{X, p}$ is regular and hence Cohen-Macaulay, there is no ambiguity about codimension as the ring is catenary, see Algebra, Lemmas 10.106.3 and 10.104.4). Hence $\dim (\mathcal{O}_{Z, p}) = \dim (\mathcal{O}_{X, p}) - r$. Choose additional $x_{r + 1}, \ldots , x_ n \in \mathfrak m_ p$ which map to a minimal system of generators of $\mathfrak m_{Z, p}$. Then $\mathfrak m_ p = (x_1, \ldots , x_ n)$ by Nakayama's lemma and we see that $D$ is a normal crossings divisor. $\square$
Lemma 41.21.3. Let $X$ be a locally Noetherian scheme. Let $D \subset X$ be a strict normal crossings divisor. If $f : Y \to X$ is a smooth morphism of schemes, then the pullback $f^*D$ is a strict normal crossings divisor on $Y$.
Proof. As $f$ is flat the pullback is defined by Divisors, Lemma 31.13.13 hence the statement makes sense. Let $q \in f^*D$ map to $p \in D$. Choose a regular system of parameters $x_1, \ldots , x_ d \in \mathfrak m_ p$ and $1 \leq r \leq d$ as in Definition 41.21.1. Since $f$ is smooth the local ring homomorphism $\mathcal{O}_{X, p} \to \mathcal{O}_{Y, q}$ is flat and the fibre ring
\[ \mathcal{O}_{Y, q}/\mathfrak m_ p \mathcal{O}_{Y, q} = \mathcal{O}_{Y_ p, q} \]
is a regular local ring (see for example Algebra, Lemma 10.140.3). Pick $y_1, \ldots , y_ n \in \mathfrak m_ q$ which map to a regular system of parameters in $\mathcal{O}_{Y_ p, q}$. Then $x_1, \ldots , x_ d, y_1, \ldots , y_ n$ generate the maximal ideal $\mathfrak m_ q$. Hence $\mathcal{O}_{Y, q}$ is a regular local ring of dimension $d + n$ by Algebra, Lemma 10.112.7 and $x_1, \ldots , x_ d, y_1, \ldots , y_ n$ is a regular system of parameters. Since $f^*D$ is cut out by $x_1 \ldots x_ r$ in $\mathcal{O}_{Y, q}$ we conclude that the lemma is true. $\square$
Here is the definition of a normal crossings divisor.
Definition 41.21.4. Let $X$ be a locally Noetherian scheme. A normal crossings divisor on $X$ is an effective Cartier divisor $D \subset X$ such that for every $p \in D$ there exists an étale morphism $U \to X$ with $p$ in the image and $D \times _ X U$ a strict normal crossings divisor on $U$.
For example $D = V(x^2 + y^2)$ is a normal crossings divisor (but not a strict one) on $\mathop{\mathrm{Spec}}(\mathbf{R}[x, y])$ because after pulling back to the étale cover $\mathop{\mathrm{Spec}}(\mathbf{C}[x, y])$ we obtain $(x - iy)(x + iy) = 0$.
Lemma 41.21.5. Let $X$ be a locally Noetherian scheme. Let $D \subset X$ be a normal crossings divisor. If $f : Y \to X$ is a smooth morphism of schemes, then the pullback $f^*D$ is a normal crossings divisor on $Y$.
Proof. As $f$ is flat the pullback is defined by Divisors, Lemma 31.13.13 hence the statement makes sense. Let $q \in f^*D$ map to $p \in D$. Choose an étale morphism $U \to X$ whose image contains $p$ such that $D \times _ X U \subset U$ is a strict normal crossings divisor as in Definition 41.21.4. Set $V = Y \times _ X U$. Then $V \to Y$ is étale as a base change of $U \to X$ (Morphisms, Lemma 29.36.4) and the pullback $D \times _ X V$ is a strict normal crossings divisor on $V$ by Lemma 41.21.3. Thus we have checked the condition of Definition 41.21.4 for $q \in f^*D$ and we conclude. $\square$
Lemma 41.21.6. Let $X$ be a locally Noetherian scheme. Let $D \subset X$ be a closed subscheme. The following are equivalent
$D$ is a normal crossings divisor in $X$,
$D$ is reduced, the normalization $\nu : D^\nu \to D$ is unramified, and for any $n \geq 1$ the scheme
is regular, the morphism $Z_ n \to X$ is a local complete intersection morphism whose conormal sheaf is locally free of rank $n$.
\[ Z_ n = D^\nu \times _ D \ldots \times _ D D^\nu \setminus \{ (p_1, \ldots , p_ n) \mid p_ i = p_ j\text{ for some }i\not= j\} \]
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. $\square$
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
$D$ is a normal crossings divisor in $X$,
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 29.54.11.
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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