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Tag 0B8B

Chapter 64: More on Morphisms of Spaces > Section 64.4: Monomorphisms

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. $\square$

    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}

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