Processing math: 100%

The Stacks project

Lemma 37.74.6. Let f : X \to Y be a separated, locally quasi-finite, and universally open morphism of schemes. Let n_{X/Y} be as in Lemma 37.74.5. If n_{X/Y}(y) \geq d for some y \in Y and d \geq 0, then n_{X/Y} \geq d in an open neighbourhood of y.

Proof. The question is local on Y hence we may assume Y affine. Let K be an algebraic closure of the residue field \kappa (y). Our assumption is that (X_ y)_ K has \geq d connected components. Then for a suitable quasi-compact open X' \subset X the scheme (X'_ y)_ K has \geq d connected components; details omitted. After replacing X by X' we may assume X is quasi-compact. Then f is quasi-finite. Let x_1, \ldots , x_ n be the points of X lying over y. Apply Lemma 37.41.5 to get an étale neighbourhood (U, u) \to (Y, y) and a decomposition

U \times _ Y X = W \amalg \ \coprod \nolimits _{i = 1, \ldots , n} \ \coprod \nolimits _{j = 1, \ldots , m_ i} V_{i, j}

as in locus citatus. Observe that n_{X/Y}(y) = \sum _ i m_ i in this situation; some details omitted. Since f is universally open, we see that V_{i, j} \to U is open for all i, j. Hence after shrinking U we may assume V_{i, j} \to U is surjective for all i, j. This proves that n_{U \times _ Y X/U} \geq \sum _ i m_ i = n_{X/Y}(y) \geq d. Since the construction of n_{X/Y} is compatible with base change the proof is complete. \square


Comments (0)


Your email address will not be published.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.