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

$D \to S$ is finite locally free,

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

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

$D \to S$ 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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