Processing math: 100%

The Stacks project

Lemma 32.20.4. Notation and assumptions as in Lemma 32.20.3. Let U \subset U' \subset X be an open containing s.

  1. Let f' : X \to U' correspond to f : X' \to U and g : Y \to \mathop{\mathrm{Spec}}(\mathcal{O}_{S, s}) via the equivalence. If f and g are separated, proper, finite, étale, then after possibly shrinking U' the morphism f' has the same property.

  2. Let a : X_1 \to X_2 be a morphism of schemes of finite presentation over U' with base change a' : X'_1 \to X'_2 over U and b : Y_1 \to Y_2 over \mathop{\mathrm{Spec}}(\mathcal{O}_{S, s}). If a' and b are separated, proper, finite, étale, then after possibly shrinking U' the morphism a has the same property.

Proof. Proof of (1). Recall that \mathop{\mathrm{Spec}}(\mathcal{O}_{S, s}) is the limit of the affine open neighbourhoods of s in S. Since g has the property in question, then the restriction of f' to one of these affine open neighbourhoods does too, see Lemmas 32.8.6, 32.13.1, 32.8.3, and 32.8.10. Since f' has the given property over U as f does, we conclude as one can check the property locally on the base.

Proof of (2). If we write \mathop{\mathrm{Spec}}(\mathcal{O}_{S, s}) = \mathop{\mathrm{lim}}\nolimits W where W runs over the affine open neighbourhoods of s in S, then we have Y_ i = \mathop{\mathrm{lim}}\nolimits W \times _ S X_ i. Thus we can use exactly the same arguments as in the proof of (1). \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Your email address will not be published.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.