The Stacks project

Lemma 35.23.3. 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:

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

  2. 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 $W(f)$. $\square$

Comments (0)

Post a comment




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 0CEY. Beware of the difference between the letter 'O' and the digit '0'.