The Stacks project

Lemma 42.16.2. Let $f : X \to Y$ be a morphism of schemes. Let $\xi \in Y$ be a point. Assume that

  1. $X$, $Y$ are integral,

  2. $Y$ is locally Noetherian

  3. $f$ is proper, dominant and $R(Y) \subset R(X)$ is finite, and

  4. $\dim (\mathcal{O}_{Y, \xi }) = 1$.

Then there exists an open neighbourhood $V \subset Y$ of $\xi $ such that $f|_{f^{-1}(V)} : f^{-1}(V) \to V$ is finite.

Proof. This lemma is a special case of Varieties, Lemma 33.17.2. Here is a direct argument in this case. By Cohomology of Schemes, Lemma 30.21.2 it suffices to prove that $f^{-1}(\{ \xi \} )$ is finite. We replace $Y$ by an affine open, say $Y = \mathop{\mathrm{Spec}}(R)$. Note that $R$ is Noetherian, as $Y$ is assumed locally Noetherian. Since $f$ is proper it is quasi-compact. Hence we can find a finite affine open covering $X = U_1 \cup \ldots \cup U_ n$ with each $U_ i = \mathop{\mathrm{Spec}}(A_ i)$. Note that $R \to A_ i$ is a finite type injective homomorphism of domains such that the induced extension of fraction fields is finite. Thus the lemma follows from Algebra, Lemma 10.113.2. $\square$

Comments (0)

There are also:

  • 2 comment(s) on Section 42.16: Preparation for principal divisors

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Your email address will not be published.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 02RM. Beware of the difference between the letter 'O' and the digit '0'.