A separation condition on an algebraic space $F$ is a condition on the diagonal morphism $F \to F \times F$. Let us first list the properties the diagonal has automatically. Since the diagonal is representable by definition the following lemma makes sense (through Definition 65.5.1).
Lemma 65.13.1. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $F$ be an algebraic space over $S$. Let $\Delta : F \to F \times F$ be the diagonal morphism. Then
$\Delta $ is locally of finite type,
$\Delta $ is a monomorphism,
$\Delta $ is separated, and
$\Delta $ is locally quasi-finite.
Proof. Let $F = U/R$ be a presentation of $F$. As in the proof of Lemma 65.10.4 the diagram
is cartesian. Hence according to Lemma 65.11.4 it suffices to show that $j$ has the properties listed in the lemma. (Note that each of the properties (1) – (4) occur in the lists of Remarks 65.4.1 and 65.4.3.) Since $j$ is an equivalence relation it is a monomorphism. Hence it is separated by Schemes, Lemma 26.23.3. As $R$ is an étale equivalence relation we see that $s, t : R \to U$ are étale. Hence $s, t$ are locally of finite type. Then it follows from Morphisms, Lemma 29.15.8 that $j$ is locally of finite type. Finally, as it is a monomorphism its fibres are finite. Thus we conclude that it is locally quasi-finite by Morphisms, Lemma 29.20.7. $\square$
Here are some common types of separation conditions, relative to the base scheme $S$. There is also an absolute notion of these conditions which we will discuss in Properties of Spaces, Section 66.3. Moreover, we will discuss separation conditions for a morphism of algebraic spaces in Morphisms of Spaces, Section 67.4.
Definition 65.13.2. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $F$ be an algebraic space over $S$. Let $\Delta : F \to F \times F$ be the diagonal morphism.
We say $F$ is separated over $S$ if $\Delta $ is a closed immersion.
We say $F$ is locally separated over $S$1 if $\Delta $ is an immersion.
We say $F$ is quasi-separated over $S$ if $\Delta $ is quasi-compact.
We say $F$ is Zariski locally quasi-separated over $S$2 if there exists a Zariski covering $F = \bigcup _{i \in I} F_ i$ such that each $F_ i$ is quasi-separated.
Note that if the diagonal is quasi-compact (when $F$ is separated or quasi-separated) then the diagonal is actually quasi-finite and separated, hence quasi-affine (by More on Morphisms, Lemma 37.43.2).
Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)