The Stacks project

Lemma 35.33.2. Let $\mathcal{P}$ be a property of morphisms of schemes which is étale local on the source-and-target. Consider the property $\mathcal{Q}$ of morphisms of germs defined by the rule

\[ \mathcal{Q}((X, x) \to (S, s)) \Leftrightarrow \text{there exists a representative }U \to S \text{ which has }\mathcal{P} \]

Then $\mathcal{Q}$ is étale local on the source-and-target as in Definition 35.33.1.

Proof. If a morphism of germs $(X, x) \to (S, s)$ has $\mathcal{Q}$, then there are arbitrarily small neighbourhoods $U \subset X$ of $x$ and $V \subset S$ of $s$ such that a representative $U \to V$ of $(X, x) \to (S, s)$ has $\mathcal{P}$. This follows from Lemma 35.32.4. Let

\[ \xymatrix{ (U', u') \ar[r]_{h'} \ar[d]_ a & (V', v') \ar[d]^ b \\ (U, u) \ar[r]^ h & (V, v) } \]

be as in Definition 35.33.1. Choose $U_1 \subset U$ and a representative $h_1 : U_1 \to V$ of $h$. Choose $V'_1 \subset V'$ and an étale representative $b_1 : V'_1 \to V$ of $b$ (Definition 35.20.2). Choose $U'_1 \subset U'$ and representatives $a_1 : U'_1 \to U_1$ and $h'_1 : U'_1 \to V'_1$ of $a$ and $h'$ with $a_1$ étale. After shrinking $U'_1$ we may assume $h_1 \circ a_1 = b_1 \circ h'_1$. By the initial remark of the proof, we are trying to show $u' \in W(h'_1) \Leftrightarrow u \in W(h_1)$ where $W(-)$ is as in Lemma 35.26.3. Thus the lemma follows from Lemma 35.32.9. $\square$

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