The Stacks project

Lemma 67.18.10. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which is locally of finite type. Let $y \in |Y|$. Let $k$ be a field and let $\mathop{\mathrm{Spec}}(k) \to Y$ be in the equivalence class defining $y$. Set $X_ k = \mathop{\mathrm{Spec}}(k) \times _ Y X$ and let $F = f^{-1}(\{ y\} )$ with the induced topology from $|X|$. Consider the following conditions

  1. $F$ is finite,

  2. $F$ is a discrete topological space,

  3. $\dim (F) = 0$,

  4. $|X_ k|$ is a finite set,

  5. $|X_ k|$ is a discrete space,

  6. $\dim (|X_ k|) = 0$,

  7. $\dim (X_ k) = 0$,

  8. $f$ is quasi-finite at all points of $|X|$ lying over $y$.

Then we have

\[ \xymatrix{ (0ACL) & (0ACP) \ar@{=>}[l] \ar@{=>}[r]_{f\text{ decent}} & (0ACQ) \ar@{<=>}[r] & (0ACR) \ar@{<=>}[r] & (0ACS) \ar@{<=>}[r] & (0ACT) } \]

If $Y$ is decent, then conditions (2) and (3) are equivalent to each other and to conditions (5), (6), (7), and (8). If $Y$ and $X$ are decent, then (1) implies all the other conditions.

Proof. By Lemma 67.18.8 conditions (5), (6), and (7) are equivalent to each other and to the condition that $X_ k \to \mathop{\mathrm{Spec}}(k)$ is locally quasi-finite. Thus by Morphisms of Spaces, Lemma 66.27.2 they are also equivalent to (8). If $f$ is decent, then $X_ k$ is a decent algebraic space and Lemma 67.18.8 shows that (4) implies (5).

The map $|X_ k| \to F$ is surjective by Properties of Spaces, Lemma 65.4.3 and we see (4) $\Rightarrow $ (1).

If $Y$ is decent, then we can pick a quasi-compact monomorphism $\mathop{\mathrm{Spec}}(k') \to Y$ in the equivalence class of $y$. In this case Lemma 67.18.6 tells us that $|X_{k'}| \to F$ is a homeomorphism. Combined with the arguments given above this implies the remaining statements of the lemma; details omitted. $\square$

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Your email address will not be published.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