# The Stacks Project

## Tag 0B89

### 67.4. Monomorphisms

This section is the continuation of Morphisms of Spaces, Section 58.10. We would like to know whether or not every monomorphism of algebraic spaces is representable. If you can prove this is true or have a counterexample, please email stacks.project@gmail.com. For the moment this is known in the following cases

1. for monomorphisms which are locally of finite type (more generally any separated, locally quasi-finite morphism is representable by Morphisms of Spaces, Lemma 58.49.1 and a monomorphism which is locally of finite type is locally quasi-finite by Morphisms of Spaces, Lemma 58.27.10),
2. if the target is a disjoint union of spectra of zero dimensional local rings (Decent Spaces, Lemma 59.18.1), and
3. for flat monomorphisms (see below).

Lemma 67.4.1 (David Rydh). A flat monomorphism of algebraic spaces is representable by schemes.

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 57.12.1), we may assume $V$ is affine. Thus we may assume $Y = \mathop{\rm Spec}(B)$ 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 $X \to X \times_{\mathop{\rm Spec}(B)} X$ is an isomorphism. Applying Limits of Spaces, Lemma 61.17.3 we reduce to the case where $B$ is local, $X \to \mathop{\rm Spec}(B)$ is a flat monomorphism, and there exists a point $x \in X$ mapping to the closed point of $\mathop{\rm Spec}(B)$. Then $X \to \mathop{\rm Spec}(B)$ is surjective as generalizations lift along flat morphisms of separated algebraic spaces, see Decent Spaces, Lemma 59.7.3. Hence we see that $\{X \to \mathop{\rm Spec}(B)\}$ is an fpqc cover. Then $X \to \mathop{\rm Spec}(B)$ is a morphism which becomes an isomorphism after base change by $X \to \mathop{\rm Spec}(B)$. Hence it is an isomorphism by fpqc descent, see Descent on Spaces, Lemma 65.10.15. $\square$

The following is (in some sense) a variant of the lemma above.

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 $Y = \mathop{\rm Spec}(A)$ is affine. Then $X$ is quasi-compact and we may choose an affine scheme $U = \mathop{\rm Spec}(B)$ and a surjective étale morphism $U \to X$ (Properties of Spaces, Lemma 57.6.3). Note that $U \times_X U = \mathop{\rm Spec}(B \otimes_A B)$. 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 57.31.1, 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 $f : X \to \mathop{\rm Spec}(A)$ 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 $\mathop{\rm Spec}(B) \to \mathop{\rm Spec}(A)$ is surjective). Hence we see that $\{X \to \mathop{\rm Spec}(A)\}$ is an fpqc cover. Then $X \to \mathop{\rm Spec}(A)$ is a morphism which becomes an isomorphism after base change by $X \to \mathop{\rm Spec}(A)$. Hence it is an isomorphism by fpqc descent, see Descent on Spaces, Lemma 65.10.15. $\square$

Lemma 67.4.3. A quasi-compact flat surjective monomorphism of algebraic spaces is an isomorphism.

Proof. Such a morphism satisfies the assumptions of Lemma 67.4.2. $\square$

The code snippet corresponding to this tag is a part of the file spaces-more-morphisms.tex and is located in lines 261–377 (see updates for more information).

\section{Monomorphisms}
\label{section-monomorphisms}

\noindent
This section is the continuation of
Morphisms of Spaces, Section \ref{spaces-morphisms-section-monomorphisms}.
We would like to know whether or not every monomorphism of algebraic
spaces is representable. If you can prove this is true or have a
\href{mailto:stacks.project@gmail.com}{stacks.project@gmail.com}.
For the moment this is known in the following cases
\begin{enumerate}
\item for monomorphisms which are locally of finite type
(more generally any separated, locally quasi-finite morphism
is representable by Morphisms of Spaces, Lemma
\ref{spaces-morphisms-lemma-locally-quasi-finite-separated-representable}
and a monomorphism which is locally of finite type is
locally quasi-finite by Morphisms of Spaces, Lemma
\ref{spaces-morphisms-lemma-monomorphism-loc-finite-type-loc-quasi-finite}),
\item if the target is a disjoint union of spectra of zero dimensional
local rings (Decent Spaces, Lemma
\ref{decent-spaces-lemma-monomorphism-toward-disjoint-union-dim-0-rings}), and
\item for flat monomorphisms (see below).
\end{enumerate}

\begin{lemma}[David Rydh]
\label{lemma-flat-case}
A flat monomorphism of algebraic spaces is representable by schemes.
\end{lemma}

\begin{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 \ref{spaces-properties-lemma-subscheme}), we may
assume $V$ is affine. Thus we may assume $Y = \Spec(B)$
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 $X \to X \times_{\Spec(B)} X$ is an isomorphism.
Applying Limits of Spaces, Lemma \ref{spaces-limits-lemma-enough-local}
we reduce to the case where $B$ is local, $X \to \Spec(B)$ is a
flat monomorphism, and
there exists a point $x \in X$ mapping to the closed point of $\Spec(B)$.
Then $X \to \Spec(B)$ is surjective as generalizations
lift along flat morphisms of separated algebraic spaces, see
Decent Spaces, Lemma \ref{decent-spaces-lemma-generalizations-lift-flat}.
Hence we see that $\{X \to \Spec(B)\}$ is an fpqc cover.
Then $X \to \Spec(B)$ is a morphism which becomes an isomorphism
after base change by $X \to \Spec(B)$. Hence it is an isomorphism by
fpqc descent, see Descent on Spaces, Lemma
\ref{spaces-descent-lemma-descending-property-isomorphism}.
\end{proof}

\noindent
The following is (in some sense) a variant of the lemma above.

\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}

\begin{lemma}
\label{lemma-flat-surjective-monomorphism}
A quasi-compact flat surjective monomorphism of algebraic spaces
is an isomorphism.
\end{lemma}

\begin{proof}
Such a morphism satisfies the assumptions of Lemma \ref{lemma-ui-case}.
\end{proof}

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