Lemma 71.6.2 (083C)—The Stacks project

Table of contents
Part 4: Algebraic Spaces
Chapter 71: Divisors on Algebraic Spaces
Section 71.6: Effective Cartier divisors
Lemma 71.6.2 (cite)

Lemma 71.6.2. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $D \subset X$ be a closed subspace. The following are equivalent:

The subspace $D$ is an effective Cartier divisor on $X$.
For some scheme $U$ and surjective étale morphism $U \to X$ the inverse image $D \times _ X U$ is an effective Cartier divisor on $U$.
For every scheme $U$ and every étale morphism $U \to X$ the inverse image $D \times _ X U$ is an effective Cartier divisor on $U$.
For every $x \in |D|$ there exists an étale morphism $(U, u) \to (X, x)$ of pointed algebraic spaces such that $U = \mathop{\mathrm{Spec}}(A)$ and $D \times _ X U = \mathop{\mathrm{Spec}}(A/(f))$ with $f \in A$ not a zerodivisor.

Proof. The equivalence of (1) – (3) follows from Definition 71.6.1 and the references preceding it. Assume (1) and let $x \in |D|$. Choose a scheme $W$ and a surjective étale morphism $W \to X$. Choose $w \in D \times _ X W$ mapping to $x$. By (3) $D \times _ X W$ is an effective Cartier divisor on $W$. Hence we can find affine étale neighbourhood $U$ by choosing an affine open neighbourhood of $w$ in $W$ as in Divisors, Lemma 31.13.2.

Assume (4). Then we see that $\mathcal{I}_ D|_ U$ is invertible by Divisors, Lemma 31.13.2. Since we can find an étale covering of $X$ by the collection of all such $U$ and $X \setminus D$, we conclude that $\mathcal{I}_ D$ is an invertible $\mathcal{O}_ X$-module. $\square$

The Stacks project

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