The Stacks project

Lemma 67.4.1. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\Delta _{X/Y} : X \to X \times _ Y X$ be the diagonal morphism. Then

  1. $\Delta _{X/Y}$ is representable,

  2. $\Delta _{X/Y}$ is locally of finite type,

  3. $\Delta _{X/Y}$ is a monomorphism,

  4. $\Delta _{X/Y}$ is separated, and

  5. $\Delta _{X/Y}$ is locally quasi-finite.

Proof. We are going to use the fact that $\Delta _{X/S}$ is representable (by definition of an algebraic space) and that it satisfies properties (2) – (5), see Spaces, Lemma 65.13.1. Note that we have a factorization

\[ X \longrightarrow X \times _ Y X \longrightarrow X \times _ S X \]

of the diagonal $\Delta _{X/S} : X \to X \times _ S X$. Since $X \times _ Y X \to X \times _ S X$ is a monomorphism, and since $\Delta _{X/S}$ is representable, it follows formally that $\Delta _{X/Y}$ is representable. In particular, the rest of the statements now make sense, see Section 67.3.

Choose a surjective étale morphism $U \to X$, with $U$ a scheme. Consider the diagram

\[ \xymatrix{ R = U \times _ X U \ar[r] \ar[d] & U \times _ Y U \ar[d] \ar[r] & U \times _ S U \ar[d] \\ X \ar[r] & X \times _ Y X \ar[r] & X \times _ S X } \]

Both squares are cartesian, hence so is the outer rectangle. The top row consists of schemes, and the vertical arrows are surjective étale morphisms. By Spaces, Lemma 65.11.4 the properties (2) – (5) for $\Delta _{X/Y}$ are equivalent to those of $R \to U \times _ Y U$. In the proof of Spaces, Lemma 65.13.1 we have seen that $R \to U \times _ S U$ has properties (2) – (5). The morphism $U \times _ Y U \to U \times _ S U$ is a monomorphism of schemes. These facts imply that $R \to U \times _ Y U$ have properties (2) – (5).

Namely: For (3), note that $R \to U \times _ Y U$ is a monomorphism as the composition $R \to U \times _ S U$ is a monomorphism. For (2), note that $R \to U \times _ Y U$ is locally of finite type, as the composition $R \to U \times _ S U$ is locally of finite type (Morphisms, Lemma 29.15.8). A monomorphism which is locally of finite type is locally quasi-finite because it has finite fibres (Morphisms, Lemma 29.20.7), hence (5). A monomorphism is separated (Schemes, Lemma 26.23.3), hence (4). $\square$

Comments (2)

Comment #875 by Matthew Emerton on

I think that should be replaced by at the end of the second paragraph.

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