The Stacks project

Lemma 44.3.2. Let $X \to S$ be a smooth morphism of schemes of relative dimension $1$. Let $D_1, D_2 \subset X$ be closed subschemes finite locally free of degrees $d_1$, $d_2$ over $S$. Then $D_1 + D_2$ is finite locally free of degree $d_1 + d_2$ over $S$.

Proof. By Lemma 44.3.1 we see that $D_1$ and $D_2$ are relative effective Cartier divisors on $X/S$. Thus $D = D_1 + D_2$ is a relative effective Cartier divisor on $X/S$ by Divisors, Lemma 31.18.3. Hence $D \to S$ is locally quasi-finite, flat, and locally of finite presentation by Lemma 44.3.1. Applying Morphisms, Lemma 29.41.11 the surjective integral morphism $D_1 \amalg D_2 \to D$ we find that $D \to S$ is separated. Then Morphisms, Lemma 29.41.9 implies that $D \to S$ is proper. This implies that $D \to S$ is finite (More on Morphisms, Lemma 37.44.1) and in turn we see that $D \to S$ is finite locally free (Morphisms, Lemma 29.48.2). Thus it suffice to show that the degree of $D \to S$ is $d_1 + d_2$. To do this we may base change to a fibre of $X \to S$, hence we may assume that $S = \mathop{\mathrm{Spec}}(k)$ for some field $k$. In this case, there exists a finite set of closed points $x_1, \ldots , x_ n \in X$ such that $D_1$ and $D_2$ are supported on $\{ x_1, \ldots , x_ n\} $. In fact, there are nonzerodivisors $f_{i, j} \in \mathcal{O}_{X, x_ i}$ such that

\[ D_1 = \coprod \mathop{\mathrm{Spec}}(\mathcal{O}_{X, x_ i}/(f_{i, 1})) \quad \text{and}\quad D_2 = \coprod \mathop{\mathrm{Spec}}(\mathcal{O}_{X, x_ i}/(f_{i, 2})) \]

Then we see that

\[ D = \coprod \mathop{\mathrm{Spec}}(\mathcal{O}_{X, x_ i}/(f_{i, 1}f_{i, 2})) \]

From this one sees easily that $D$ has degree $d_1 + d_2$ over $k$ (if need be, use Algebra, Lemma 10.121.1). $\square$

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