The Stacks project

Lemma 70.6.2. With notation and assumptions as in Situation 70.6.1. If

  1. $f$ is étale,

  2. $f_0$ is locally of finite presentation,

then $f_ i$ is étale for some $i \geq 0$.

Proof. Choose an affine scheme $V_0$ and a surjective étale morphism $V_0 \to Y_0$. Choose an affine scheme $U_0$ and a surjective étale morphism $U_0 \to V_0 \times _{Y_0} X_0$. Diagram

\[ \xymatrix{ U_0 \ar[d] \ar[r] & V_0 \ar[d] \\ X_0 \ar[r] & Y_0 } \]

The vertical arrows are surjective and étale by construction. We can base change this diagram to $B_ i$ or $B$ to get

\[ \vcenter { \xymatrix{ U_ i \ar[d] \ar[r] & V_ i \ar[d] \\ X_ i \ar[r] & Y_ i } } \quad \text{and}\quad \vcenter { \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } } \]

Note that $U_ i, V_ i, U, V$ are affine schemes, the vertical morphisms are surjective étale, and the limit of the morphisms $U_ i \to V_ i$ is $U \to V$. Recall that $X_ i \to Y_ i$ is étale if and only if $U_ i \to V_ i$ is étale and similarly $X \to Y$ is étale if and only if $U \to V$ is étale (Morphisms of Spaces, Lemma 67.39.2). Since $f_0$ is locally of finite presentation, so is the morphism $U_0 \to V_0$. Hence the lemma follows from Limits, Lemma 32.8.10. $\square$

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Your email address will not be published.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 07SL. Beware of the difference between the letter 'O' and the digit '0'.