The Stacks project

Lemma 65.12.2. Let $S$ be a scheme. Let $j : R \to U \times _ S U$ be an étale equivalence relation. Let $X = U/R$ be the associated algebraic space (Spaces, Theorem 64.10.5). There is a canonical bijection

\[ R\text{-invariant locally closed subschemes }Z'\text{ of }U \leftrightarrow \text{locally closed subspaces }Z\text{ of }X \]

Moreover, if $Z \to X$ is closed (resp. open) if and only if $Z' \to U$ is closed (resp. open).

Proof. Denote $\varphi : U \to X$ the canonical map. The bijection sends $Z \to X$ to $Z' = Z \times _ X U \to U$. It is immediate from the definition that $Z' \to U$ is an immersion, resp. closed immersion, resp. open immersion if $Z \to X$ is so. It is also clear that $Z'$ is $R$-invariant (see Groupoids, Definition 39.19.1).

Conversely, assume that $Z' \to U$ is an immersion which is $R$-invariant. Let $R'$ be the restriction of $R$ to $Z'$, see Groupoids, Definition 39.18.2. Since $R' = R \times _{s, U} Z' = Z' \times _{U, t} R$ in this case we see that $R'$ is an étale equivalence relation on $Z'$. By Spaces, Theorem 64.10.5 we see $Z = Z'/R'$ is an algebraic space. By construction we have $U \times _ X Z = Z'$, so $U \times _ X Z \to Z$ is an immersion. Note that the property “immersion” is preserved under base change and fppf local on the base (see Spaces, Section 64.4). Moreover, immersions are separated and locally quasi-finite (see Schemes, Lemma 26.23.8 and Morphisms, Lemma 29.20.16). Hence by More on Morphisms, Lemma 37.55.1 immersions satisfy descent for fppf covering. This means all the hypotheses of Spaces, Lemma 64.11.5 are satisfied for $Z \to X$, $\mathcal{P}=$“immersion”, and the étale surjective morphism $U \to X$. We conclude that $Z \to X$ is representable and an immersion, which is the definition of a subspace (see Spaces, Definition 64.12.1).

It is clear that these constructions are inverse to each other and we win. $\square$

Comments (0)

There are also:

  • 2 comment(s) on Section 65.12: Reduced spaces

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Your email address will not be published.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 07TW. Beware of the difference between the letter 'O' and the digit '0'.