The Stacks project

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 67.39.7 and 67.39.8. By Lemma 80.3.8 $U \to G$ 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 80.4.2). 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 67.50.2 we conclude that $E$ is a scheme also. By Descent, Lemma 35.39.1 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 37.57.1 this descent datum is effective. By Descent, Lemma 35.39.1 again this implies that $H$ is a scheme. By Descent, Lemmas 35.23.6 and 35.23.24 it now follows that $b$ is separated and locally quasi-finite. $\square$