The Stacks project

Proof. Since $M$ represents the quotient sheaf we have a cartesian diagram

of schemes. Since $U \times _ S U \to M \times _ S M$ is surjective finite locally free, to show that $M \to M \times _ S M$ is quasi-compact, resp. a closed immersion, it suffices to show that $j : R \to U \times _ S U$ is quasi-compact, resp. a closed immersion, see Descent, Lemmas 35.23.1 and 35.23.19. Since $j : R \to U \times _ S U$ is a morphism over $U$ and since $R$ is finite over $U$, we see that $j$ is quasi-compact as soon as the projection $U \times _ S U \to U$ is quasi-separated (Schemes, Lemma 26.21.14). Since $j$ is a monomorphism and locally of finite type, we see that $j$ is a closed immersion as soon as it is proper (Étale Morphisms, Lemma 41.7.2) which will be the case as soon as the projection $U \times _ S U \to U$ is separated (Morphisms, Lemma 29.41.7). This proves (1) and (3). To prove (2) and (4) we replace $S$ by $\mathop{\mathrm{Spec}}(\mathbf{Z})$, see Definition 66.3.1. Since Lemma 66.14.2 is proved through an application of Proposition 66.14.1 the final statement is clear too. $\square$