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Tag 0B89

64.4. Monomorphisms

This section is the continuation of Morphisms of Spaces, Section 55.10. 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 55.49.1 and a monomorphism which is locally of finite type is locally quasi-finite by Morphisms of Spaces, Lemma 55.27.10),
  2. if the target is a disjoint union of spectra of zero dimensional local rings (Decent Spaces, Lemma 56.18.1), and
  3. for flat monomorphisms (see below).

Lemma 64.4.1 (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 54.12.1), 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 58.17.3 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 56.7.3. 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 62.10.15. $\square$

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

Lemma 64.4.2. 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 54.6.3). 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 54.31.1, 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 62.10.15. $\square$

Lemma 64.4.3. A quasi-compact flat surjective monomorphism of algebraic spaces is an isomorphism.

Proof. Such a morphism satisfies the assumptions of Lemma 64.4.2. $\square$

    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
    counterexample, please email
    \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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