Lemma 107.5.20. If $\mathcal{X}$ is a locally Noetherian algebraic stack, and if $x \in |\mathcal{X}|$, then for any open substack $\mathcal{V}$ of $\mathcal{X}$ containing $x$, there is a finite type point $x_0 \in |\mathcal{V}|$ such that $\dim _{x_0}(\mathcal{X}) = \dim _ x(\mathcal{V})$.

**Proof.**
Choose a smooth surjective morphism $f:U \to \mathcal{X}$ whose source is a scheme, and consider the function $u \mapsto \dim _{f(u)}(\mathcal{X});$ since the morphism $|U| \to |\mathcal{X}|$ induced by $f$ is open (as $f$ is smooth) as well as surjective (by assumption), and takes finite type points to finite type points (by the very definition of the finite type points of $|\mathcal{X}|$), it suffices to show that for any $u \in U$, and any open neighbourhood of $u$, there is a finite type point $u_0$ in this neighbourhood such that $\dim _{f(u_0)}(\mathcal{X}) = \dim _{f(u)}(\mathcal{X}).$ Since, with this reformulation of the problem, the surjectivity of $f$ is no longer required, we may replace $U$ by the open neighbourhood of the point $u$ in question, and thus reduce to the problem of showing that for each $u \in U$, there is a finite type point $u_0 \in U$ such that $\dim _{f(u_0)}(\mathcal{X}) = \dim _{f(u)}(\mathcal{X}).$ By Lemma 107.5.4 $\dim _{f(u)}(\mathcal{X}) = \dim _ u(U) - \dim _ u(U_{f(u)}),$ while $\dim _{f(u_0)}(\mathcal{X}) = \dim _{u_0}(U) - \dim _{u_0}(U_{f(u_0)}).$ Since $f$ is smooth, the expression $\dim _{u_0}(U_{f(u_0)})$ is locally constant as $u_0$ varies over $U$ (by Lemma 107.5.11 (2)), and so shrinking $U$ further around $u$ if necessary, we may assume it is constant. Thus the problem becomes to show that we may find a finite type point $u_0 \in U$ for which $\dim _{u_0}(U) = \dim _ u(U)$. Since by definition $\dim _ u U$ is the minimum of the dimensions $\dim V$, as $V$ ranges over the open neighbourhoods $V$ of $u$ in $U$, we may shrink $U$ down further around $u$ so that $\dim _ u U = \dim U$. The existence of desired point $u_0$ then follows from Lemma 107.5.12.
$\square$

## 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Your email address will not be published.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)