Lemma 87.11.2 (0AIP)—The Stacks project

Lemma 87.11.2. Let $S$ be a scheme. If $X$ is a formal algebraic space over $S$, then the diagonal morphism $\Delta : X \to X \times _ S X$ is representable, a monomorphism, locally quasi-finite, locally of finite type, and separated.

Proof. Suppose given $U \to X$ and $V \to X$ with $U, V$ schemes over $S$. Then $U \times _ X V$ is a sheaf. Choose $\{ X_ i \to X\} $ as in Definition 87.11.1. For every $i$ the morphism

\[ (U \times _ X X_ i) \times _{X_ i} (V \times _ X X_ i) = (U \times _ X V) \times _ X X_ i \to U \times _ X V \]

is representable and étale as a base change of $X_ i \to X$ and its source is a scheme (use Lemmas 87.9.2 and 87.9.11). These maps are jointly surjective hence $U \times _ X V$ is an algebraic space by Bootstrap, Theorem 80.10.1. The morphism $U \times _ X V \to U \times _ S V$ is a monomorphism. It is also locally quasi-finite, because on precomposing with the morphism displayed above we obtain the composition

\[ (U \times _ X X_ i) \times _{X_ i} (V \times _ X X_ i) \to (U \times _ X X_ i) \times _ S (V \times _ X X_ i) \to U \times _ S V \]

which is locally quasi-finite as a composition of a closed immersion (Lemma 87.9.2) and an étale morphism, see Descent on Spaces, Lemma 74.19.2. Hence we conclude that $U \times _ X V$ is a scheme by Morphisms of Spaces, Proposition 67.50.2. Thus $\Delta $ is representable, see Spaces, Lemma 65.5.10.

In fact, since we've shown above that the morphisms of schemes $U \times _ X V \to U \times _ S V$ are always monomorphisms and locally quasi-finite we conclude that $\Delta : X \to X \times _ S X$ is a monomorphism and locally quasi-finite, see Spaces, Lemma 65.5.11. Then we can use the principle of Spaces, Lemma 65.5.8 to see that $\Delta $ is separated and locally of finite type. Namely, a monomorphism of schemes is separated (Schemes, Lemma 26.23.3) and a locally quasi-finite morphism of schemes is locally of finite type (follows from the definition in Morphisms, Section 29.20). $\square$

Comments (3)

Comment #1944 by Brian Conrad on April 30, 2016 at 16:46

The last paragraph is re-inventing the wheel: any $S$ -morphism $f:W \rightarrow X \times_S X$ factors as $W \rightarrow W \times_S W \rightarrow X \times_S W$ where the first step is $\Delta_{W/S}$ and the second is $f_1 \times f_2$ for $f_j = {\rm{pr}}_j \circ f$ , so the settled study of pairs of $S$ -maps $U, V \rightrightarrows X$ yields the desired properties of $f$ instantly via base change along $\Delta_{W/S}$ without needing to redo anything. This same type of formal argument applies to many other situations with proving things about algebraic spaces, and I think the same trick is used in Knutson's book on algebraic spaces.

Also, in the statement of the Lemma it is redundant to include "locally of finite type" since that is part of "locally quasi-finite", and the proof directly establishes locally quasi-finiteness without first proving locally of finite type.

Comment #1945 by Brian Conrad on April 30, 2016 at 16:48

Well, in the final paragraph you do note that separatedness is a consequence of being a monomorphism, which had not been remarked earlier. So that part of the final paragraph can be retained. The rest could be disposed of by referring to some super-general and easy Lemma about manipulating properties of morphisms via the trick mentioned in my preceding comment here.

Comment #2001 by Johan on May 04, 2016 at 11:01

Thanks. I have made the changes you suggested here except that I have not removed the property "locally of finite type". We can do this later, but then we would have to do this everywhere because I've consistently added this property to the list of properties of diagonals of algebraic spaces. We could also remove "separated" as it is implied by monomorphism. My experience is that it is sometimes useful to state things which are obviously implied for use in later lemmas, but I agree this isn't a strong argument.

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