The Stacks project

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

  1. $D$ is a strict normal crossings divisor, and

  2. $D$ 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. $\square$

Comments (2)

Comment #2649 by David Hansen on

In condition (2) of the Lemma, you need to allow for the possibility that some D_J's are empty.

Comment #2668 by on

OK David, I know we talked this over, but now that I look at it more closely, I see that the formulation is correct. Namely, it says that every irreducible component should have the correct codimension. So if is empty, then (a) it is regular, and (b) there is no irreducible component so the condition on codimension is void. So it is OK as formulated.

Also everybody: please note that we have only defined the codimension of irreducible closed subsets in 5.11. I strongly suggest never thinking about codimensions of arbitrary closed subsets of topological spaces...

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