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63.13 Separation conditions on algebraic spaces

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 63.5.1).

Lemma 63.13.1. 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

  1. $\Delta $ is locally of finite type,

  2. $\Delta $ is a monomorphism,

  3. $\Delta $ is separated, and

  4. $\Delta $ is locally quasi-finite.

Proof. Let $F = U/R$ be a presentation of $F$. As in the proof of Lemma 63.10.4 the diagram

\[ \xymatrix{ R \ar[r] \ar[d]_ j & F \ar[d]^\Delta \\ U \times _ S U \ar[r] & F \times F } \]

is cartesian. Hence according to Lemma 63.11.4 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 63.4.1 and 63.4.3.) 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 64.3. Moreover, we will discuss separation conditions for a morphism of algebraic spaces in Morphisms of Spaces, Section 65.4.

Definition 63.13.2. 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.

  1. We say $F$ is separated over $S$ if $\Delta $ is a closed immersion.

  2. We say $F$ is locally separated over $S$1 if $\Delta $ is an immersion.

  3. We say $F$ is quasi-separated over $S$ if $\Delta $ is quasi-compact.

  4. 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 37.39.2).

[1] In the literature this often refers to quasi-separated and locally separated algebraic spaces.
[2] This definition was suggested by B. Conrad.

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