The Stacks project

Lemma 37.61.4. Let $f : X \to S$ be a morphism of schemes. The following are equivalent

  1. The morphism $f$ is weakly étale.

  2. For every affine opens $U \subset X$, $V \subset S$ with $f(U) \subset V$ the ring map $\mathcal{O}_ S(V) \to \mathcal{O}_ X(U)$ is weakly étale.

  3. There exists an open covering $S = \bigcup _{j \in J} V_ j$ and open coverings $f^{-1}(V_ j) = \bigcup _{i \in I_ j} U_ i$ such that each of the morphisms $U_ i \to V_ j$, $j\in J, i\in I_ j$ is weakly étale.

  4. There exists an affine open covering $S = \bigcup _{j \in J} V_ j$ and affine open coverings $f^{-1}(V_ j) = \bigcup _{i \in I_ j} U_ i$ such that the ring map $\mathcal{O}_ S(V_ j) \to \mathcal{O}_ X(U_ i)$ is of weakly étale, for all $j\in J, i\in I_ j$.

Moreover, if $f$ is weakly étale then for any open subschemes $U \subset X$, $V \subset S$ with $f(U) \subset V$ the restriction $f|_ U : U \to V$ is weakly-étale.

Proof. Suppose given open subschemes $U \subset X$, $V \subset S$ with $f(U) \subset V$. Then $U \times _ V U \subset X \times _ Y X$ is open (Schemes, Lemma 26.17.3) and the diagonal $\Delta _{U/V}$ of $f|_ U : U \to V$ is the restriction $\Delta _{X/Y}|_ U : U \to U \times _ V U$. Since flatness is a local property of morphisms of schemes (Morphisms, Lemma 29.25.3) the final statement of the lemma is follows as well as the equivalence of (1) and (3). If $X$ and $Y$ are affine, then $X \to Y$ is weakly étale if and only if $\mathcal{O}_ Y(Y) \to \mathcal{O}_ X(X)$ is weakly étale (use again Morphisms, Lemma 29.25.3). Thus (1) and (3) are also equivalent to (2) and (4). $\square$

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Your email address will not be published.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 094S. Beware of the difference between the letter 'O' and the digit '0'.