# The Stacks Project

## Tag 0B8B

Lemma 67.4.2. Let $S$ be a scheme. Let $f : X \to Y$ be a quasi-compact monomorphism of algebraic spaces $f : X \to Y$ such that for every $T \to X$ 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 57.6.3). 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 57.32.1, Descent, Definition 34.3.1, and Descent, Subsection 34.4.14. 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 34.4.22. 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 65.10.15. $\square$

The code snippet corresponding to this tag is a part of the file spaces-more-morphisms.tex and is located in lines 318–326 (see updates for more information).

\begin{lemma}
\label{lemma-ui-case}
Let $S$ be a scheme. Let $f : X \to Y$ be a quasi-compact monomorphism
of algebraic spaces $f : X \to Y$ such that for every $T \to X$ 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).
\end{lemma}

\begin{proof}
The question is \'etale local on $Y$, hence we may assume $Y = \Spec(A)$
is affine. Then $X$ is quasi-compact and we may choose an affine scheme
$U = \Spec(B)$ and a surjective \'etale morphism $U \to X$
(Properties of Spaces, Lemma
\ref{spaces-properties-lemma-quasi-compact-affine-cover}).
Note that $U \times_X U = \Spec(B \otimes_A B)$. 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
\ref{spaces-properties-proposition-quasi-coherent},
Descent, Definition \ref{descent-definition-descent-datum-modules}, and
Descent, Subsection \ref{descent-subsection-descent-modules-morphisms}.
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 \ref{descent-theorem-descent}. Thus pullback along
$f : X \to \Spec(A)$ 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 $\Spec(B) \to \Spec(A)$ is surjective).
Hence we see that $\{X \to \Spec(A)\}$ is an fpqc cover.
Then $X \to \Spec(A)$ is a morphism which becomes an isomorphism
after base change by $X \to \Spec(A)$. Hence it is an isomorphism by
fpqc descent, see Descent on Spaces, Lemma
\ref{spaces-descent-lemma-descending-property-isomorphism}.
\end{proof}

## Comments (0)

There are no comments yet for this tag.

## Add a comment on tag 0B8B

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the lower-right corner).

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following box. So in case this where tag 0321 you just have to write 0321. Beware of the difference between the letter 'O' and the digit 0.

This captcha seems more appropriate than the usual illegible gibberish, right?