The Stacks project

Lemma 35.30.2. 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 35.30.1.

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 35.29.4. 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 35.30.1. 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 35.17.2). 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 35.23.3. Thus the lemma follows from Lemma 35.29.9. $\square$


Comments (0)


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Your email address will not be published.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 04R6. Beware of the difference between the letter 'O' and the digit '0'.