The Stacks project

Lemma 76.4.2. Let $S$ be a scheme. Let $f : X \to Y$ be a quasi-compact monomorphism of algebraic spaces such that for every $T \to Y$ the map

\[ \mathcal{O}_ T \to f_{T,*}\mathcal{O}_{X \times _ Y T} \]

is injective. Then $f$ is an isomorphism (and hence representable by schemes).

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,

\[ A \to B \]

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. $\square$


Comments (3)

Comment #5008 by Laurent Moret-Bailly on

Typo in statement: "for every " should be "for every ".

Comment #5009 by Matthieu Romagny on

And is duplicated.


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