77.5 Bootstrapping the diagonal

Lemma 77.5.1. Let $S$ be a scheme. If $F$ is a presheaf on $(\mathit{Sch}/S)_{fppf}$. The following are equivalent:

1. $\Delta _ F : F \to F \times F$ is representable by algebraic spaces,

2. for every scheme $T$ any map $T \to F$ is representable by algebraic spaces, and

3. for every algebraic space $X$ any map $X \to F$ is representable by algebraic spaces.

Proof. Assume (1). Let $X \to F$ be as in (3). Let $T$ be a scheme, and let $T \to F$ be a morphism. Then we have

$T \times _ F X = (T \times _ S X) \times _{F \times F, \Delta } F$

which is an algebraic space by Lemma 77.3.7 and (1). Hence $X \to F$ is representable, i.e., (3) holds. The implication (3) $\Rightarrow$ (2) is trivial. Assume (2). Let $T$ be a scheme, and let $(a, b) : T \to F \times F$ be a morphism. Then

$F \times _{\Delta _ F, F \times F} T = T \times _{a, F, b} T$

which is an algebraic space by assumption. Hence $\Delta _ F$ is representable by algebraic spaces, i.e., (1) holds. $\square$

In particular if $F$ is a presheaf satisfying the equivalent conditions of the lemma, then for any morphism $X \to F$ where $X$ is an algebraic space it makes sense to say that $X \to F$ is surjective (resp. étale, flat, locally of finite presentation) by using Definition 77.4.1.

Before we actually do the bootstrap we prove a fun lemma.

Lemma 77.5.2. Let $S$ be a scheme. Let

$\xymatrix{ E \ar[r]_ a \ar[d]_ f & F \ar[d]^ g \\ H \ar[r]^ b & G }$

be a cartesian diagram of sheaves on $(\mathit{Sch}/S)_{fppf}$, so $E = H \times _ G F$. If

1. $g$ is representable by algebraic spaces, surjective, flat, and locally of finite presentation, and

2. $a$ is representable by algebraic spaces, separated, and locally quasi-finite

then $b$ is representable (by schemes) as well as separated and locally quasi-finite.

Proof. Let $T$ be a scheme, and let $T \to G$ be a morphism. We have to show that $T \times _ G H$ is a scheme, and that the morphism $T \times _ G H \to T$ is separated and locally quasi-finite. Thus we may base change the whole diagram to $T$ and assume that $G$ is a scheme. In this case $F$ is an algebraic space. Let $U$ be a scheme, and let $U \to F$ be a surjective étale morphism. Then $U \to F$ is representable, surjective, flat and locally of finite presentation by Morphisms of Spaces, Lemmas 64.39.7 and 64.39.8. By Lemma 77.3.8 $U \to G$ is surjective, flat and locally of finite presentation also. Note that the base change $E \times _ F U \to U$ of $a$ is still separated and locally quasi-finite (by Lemma 77.4.2). Hence we may replace the upper part of the diagram of the lemma by $E \times _ F U \to U$. In other words, we may assume that $F \to G$ is a surjective, flat morphism of schemes which is locally of finite presentation. In particular, $\{ F \to G\}$ is an fppf covering of schemes. By Morphisms of Spaces, Proposition 64.50.2 we conclude that $E$ is a scheme also. By Descent, Lemma 35.36.1 the fact that $E = H \times _ G F$ means that we get a descent datum on $E$ relative to the fppf covering $\{ F \to G\}$. By More on Morphisms, Lemma 37.49.1 this descent datum is effective. By Descent, Lemma 35.36.1 again this implies that $H$ is a scheme. By Descent, Lemmas 35.20.6 and 35.20.24 it now follows that $b$ is separated and locally quasi-finite. $\square$

Here is the result that the section title refers to.

Lemma 77.5.3. Let $S$ be a scheme. Let $F : (\mathit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}$ be a functor. Assume that

1. the presheaf $F$ is a sheaf,

2. there exists an algebraic space $X$ and a map $X \to F$ which is representable by algebraic spaces, surjective, flat and locally of finite presentation.

Then $\Delta _ F$ is representable (by schemes).

Proof. Let $U \to X$ be a surjective étale morphism from a scheme towards $X$. Then $U \to X$ is representable, surjective, flat and locally of finite presentation by Morphisms of Spaces, Lemmas 64.39.7 and 64.39.8. By Lemma 77.4.3 the composition $U \to F$ is representable by algebraic spaces, surjective, flat and locally of finite presentation also. Thus we see that $R = U \times _ F U$ is an algebraic space, see Lemma 77.3.7. The morphism of algebraic spaces $R \to U \times _ S U$ is a monomorphism, hence separated (as the diagonal of a monomorphism is an isomorphism, see Morphisms of Spaces, Lemma 64.10.2). Since $U \to F$ is locally of finite presentation, both morphisms $R \to U$ are locally of finite presentation, see Lemma 77.4.2. Hence $R \to U \times _ S U$ is locally of finite type (use Morphisms of Spaces, Lemmas 64.28.5 and 64.23.6). Altogether this means that $R \to U \times _ S U$ is a monomorphism which is locally of finite type, hence a separated and locally quasi-finite morphism, see Morphisms of Spaces, Lemma 64.27.10.

Now we are ready to prove that $\Delta _ F$ is representable. Let $T$ be a scheme, and let $(a, b) : T \to F \times F$ be a morphism. Set

$T' = (U \times _ S U) \times _{F \times F} T.$

Note that $U \times _ S U \to F \times F$ is representable by algebraic spaces, surjective, flat and locally of finite presentation by Lemma 77.4.4. Hence $T'$ is an algebraic space, and the projection morphism $T' \to T$ is surjective, flat, and locally of finite presentation. Consider $Z = T \times _{F \times F} F$ (this is a sheaf) and

$Z' = T' \times _{U \times _ S U} R = T' \times _ T Z.$

We see that $Z'$ is an algebraic space, and $Z' \to T'$ is separated and locally quasi-finite by the discussion in the first paragraph of the proof which showed that $R$ is an algebraic space and that the morphism $R \to U \times _ S U$ has those properties. Hence we may apply Lemma 77.5.2 to the diagram

$\xymatrix{ Z' \ar[r] \ar[d] & T' \ar[d] \\ Z \ar[r] & T }$

and we conclude. $\square$

Here is a variant of the result above.

Lemma 77.5.4. Let $S$ be a scheme. Let $F : (\mathit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}$ be a functor. Let $X$ be a scheme and let $X \to F$ be representable by algebraic spaces and locally quasi-finite. Then $X \to F$ is representable (by schemes).

Proof. Let $T$ be a scheme and let $T \to F$ be a morphism. We have to show that the algebraic space $X \times _ F T$ is representable by a scheme. Consider the morphism

$X \times _ F T \longrightarrow X \times _{\mathop{\mathrm{Spec}}(\mathbf{Z})} T$

Since $X \times _ F T \to T$ is locally quasi-finite, so is the displayed arrow (Morphisms of Spaces, Lemma 64.27.8). On the other hand, the displayed arrow is a monomorphism and hence separated (Morphisms of Spaces, Lemma 64.10.3). Thus $X \times _ F T$ is a scheme by Morphisms of Spaces, Proposition 64.50.2. $\square$

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