The Stacks project

Lemma 44.3.1. Let $X \to S$ be a smooth morphism of schemes of relative dimension $1$. Let $D \subset X$ be a closed subscheme. Consider the following conditions

  1. $D \to S$ is finite locally free,

  2. $D$ is a relative effective Cartier divisor on $X/S$,

  3. $D \to S$ is locally quasi-finite, flat, and locally of finite presentation, and

  4. $D \to S$ is locally quasi-finite and flat.

We always have the implications

\[ (1) \Rightarrow (2) \Leftrightarrow (3) \Rightarrow (4) \]

If $S$ is locally Noetherian, then the last arrow is an if and only if. If $X \to S$ is proper (and $S$ arbitrary), then the first arrow is an if and only if.

Proof. Equivalence of (2) and (3). This follows from Divisors, Lemma 31.18.9 if we can show the equivalence of (2) and (3) when $S$ is the spectrum of a field $k$. Let $x \in X$ be a closed point. As $X$ is smooth of relative dimension $1$ over $k$ and we see that $\mathcal{O}_{X, x}$ is a regular local ring of dimension $1$ (see Varieties, Lemma 33.25.3). Thus $\mathcal{O}_{X, x}$ is a discrete valuation ring (Algebra, Lemma 10.119.7) and hence a PID. It follows that every sheaf of ideals $\mathcal{I} \subset \mathcal{O}_ X$ which is nonvanishing at all the generic points of $X$ is invertible (Divisors, Lemma 31.15.2). In other words, every closed subscheme of $X$ which does not contain a generic point is an effective Cartier divisor. It follows that (2) and (3) are equivalent.

If $S$ is Noetherian, then any locally quasi-finite morphism $D \to S$ is locally of finite presentation (Morphisms, Lemma 29.21.9), whence (3) is equivalent to (4).

If $X \to S$ is proper (and $S$ is arbitrary), then $D \to S$ is proper as well. Since a proper locally quasi-finite morphism is finite (More on Morphisms, Lemma 37.44.1) and a finite, flat, and finitely presented morphism is finite locally free (Morphisms, Lemma 29.48.2), we see that (1) is equivalent to (2). $\square$

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