The Stacks project

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 ├ętale 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. $\square$


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