Lemma 32.20.4 (0EY3)—The Stacks project

Table of contents
Part 2: Schemes
Chapter 32: Limits of Schemes
Section 32.20: Glueing in closed fibres
Lemma 32.20.4 (cite)

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

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.
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). $\square$

The Stacks project

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