Lemma 35.26.3. Let $\tau \in \{ fppf, syntomic, smooth, {\acute{e}tale}\} $. Let $\mathcal{P}$ be a property of morphisms which is $\tau $ local on the source. For any morphism of schemes $f : X \to Y$ there exists a largest open $W(f) \subset X$ such that the restriction $f|_{W(f)} : W(f) \to Y$ has $\mathcal{P}$. Moreover, if $g : X' \to X$ is flat and locally of finite presentation, syntomic, smooth, or étale and $f' = f \circ g : X' \to Y$, then $g^{-1}(W(f)) = W(f')$.

**Proof.**
Consider the union $W$ of the images $g(X') \subset X$ of morphisms $g : X' \to X$ with the properties:

$g$ is flat and locally of finite presentation, syntomic, smooth, or étale, and

the composition $X' \to X \to Y$ has property $\mathcal{P}$.

Since such a morphism $g$ is open (see Morphisms, Lemma 29.25.10) we see that $W \subset X$ is an open subset of $X$. Since $\mathcal{P}$ is local in the $\tau $ topology the restriction $f|_ W : W \to Y$ has property $\mathcal{P}$ because we are given a $\tau $ covering $\{ X' \to W\} $ of $W$ such that the pullbacks have $\mathcal{P}$. This proves the existence of $W(f)$. The compatibility stated in the last sentence follows immediately from the construction of $W(f)$. $\square$

## 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Your email address will not be published.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.

## Comments (0)