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 37.40.1) 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). $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0B9E. Beware of the difference between the letter 'O' and the digit '0'.