The Stacks project

Proof. Let $f : X \to Y$ be a flat monomorphism of algebraic spaces. To prove $f$ is representable, we have to show $X \times _ Y V$ is a scheme for every scheme $V$ mapping to $Y$. Since being a scheme is local (Properties of Spaces, Lemma 66.13.1), we may assume $V$ is affine. Thus we may assume $Y = \mathop{\mathrm{Spec}}(B)$ is an affine scheme. Next, we can assume that $X$ is quasi-compact by replacing $X$ by a quasi-compact open. The space $X$ is separated as $X \to X \times _{\mathop{\mathrm{Spec}}(B)} X$ is an isomorphism. Applying Limits of Spaces, Lemma 70.17.3 we reduce to the case where $B$ is local, $X \to \mathop{\mathrm{Spec}}(B)$ is a flat monomorphism, and there exists a point $x \in X$ mapping to the closed point of $\mathop{\mathrm{Spec}}(B)$. Then $X \to \mathop{\mathrm{Spec}}(B)$ is surjective as generalizations lift along flat morphisms of separated algebraic spaces, see Decent Spaces, Lemma 68.7.4. Hence we see that $\{ X \to \mathop{\mathrm{Spec}}(B)\} $ is an fpqc cover. Then $X \to \mathop{\mathrm{Spec}}(B)$ is a morphism which becomes an isomorphism after base change by $X \to \mathop{\mathrm{Spec}}(B)$. Hence it is an isomorphism by fpqc descent, see Descent on Spaces, Lemma 74.11.15. $\square$

Proof. The question is étale local on $Y$, hence we may assume $Y = \mathop{\mathrm{Spec}}(A)$ is affine. Then $X$ is quasi-compact and we may choose an affine scheme $U = \mathop{\mathrm{Spec}}(B)$ and a surjective étale morphism $U \to X$ (Properties of Spaces, Lemma 66.6.3). Note that $U \times _ X U = \mathop{\mathrm{Spec}}(B \otimes _ A B)$. Hence the category of quasi-coherent $\mathcal{O}_ X$-modules is equivalent to the category $DD_{B/A}$ of descent data on modules for $A \to B$. See Properties of Spaces, Proposition 66.32.1, Descent, Definition 35.3.1, and Descent, Subsection 35.4.14. On the other hand,

is a universally injective ring map. Namely, given an $A$-module $M$ we see that $A \oplus M \to B \otimes _ A (A \oplus M)$ is injective by the assumption of the lemma. Hence $DD_{B/A}$ is equivalent to the category of $A$-modules by Descent, Theorem 35.4.22. Thus pullback along $f : X \to \mathop{\mathrm{Spec}}(A)$ determines an equivalence of categories of quasi-coherent modules. In particular $f^*$ is exact on quasi-coherent modules and we see that $f$ is flat (small detail omitted). Moreover, it is clear that $f$ is surjective (for example because $\mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)$ is surjective). Hence we see that $\{ X \to \mathop{\mathrm{Spec}}(A)\} $ is an fpqc cover. Then $X \to \mathop{\mathrm{Spec}}(A)$ is a morphism which becomes an isomorphism after base change by $X \to \mathop{\mathrm{Spec}}(A)$. Hence it is an isomorphism by fpqc descent, see Descent on Spaces, Lemma 74.11.15. $\square$

Proof. Such a morphism satisfies the assumptions of Lemma 76.4.2. $\square$