The Stacks project

Lemma 67.34.4. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $n \geq 0$. Assume $f$ is locally of finite type. The set

\[ W_ n = \{ x \in |X| \text{ such that the relative dimension of }f\text{ at } x \leq n\} \]

is open in $|X|$.

Proof. Choose a diagram

\[ \xymatrix{ U \ar[r]_ h \ar[d]_ a & V \ar[d] \\ X \ar[r] & Y } \]

where $U$ and $V$ are schemes and the vertical arrows are surjective and ├ętale, see Spaces, Lemma 65.11.6. By Morphisms, Lemma 29.28.4 the set $U_ n$ of points where $h$ has relative dimension $\leq n$ is open in $U$. By our definition of relative dimension for morphisms of algebraic spaces at points we see that $U_ n = a^{-1}(W_ n)$. The lemma follows by definition of the topology on $|X|$. $\square$

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Your email address will not be published.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