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Tag 01R8

Chapter 28: Morphisms of Schemes > Section 28.6: Scheme theoretic image

Lemma 28.6.3. Let $f : X \to Y$ be a morphism of schemes. Let $Z \subset Y$ be the scheme theoretic image of $f$. If $f$ is quasi-compact then

  1. the sheaf of ideals $\mathcal{I} = \mathop{\rm Ker}(\mathcal{O}_Y \to f_*\mathcal{O}_X)$ is quasi-coherent,
  2. the scheme theoretic image $Z$ is the closed subscheme determined by $\mathcal{I}$,
  3. for any open $U \subset Y$ the scheme theoretic image of $f|_{f^{-1}(U)} : f^{-1}(U) \to U$ is equal to $Z \cap U$, and
  4. the image $f(X) \subset Z$ is a dense subset of $Z$, in other words the morphism $X \to Z$ is dominant (see Definition 28.8.1).

Proof. Part (4) follows from part (3). To show (3) it suffices to prove (1) since the formation of $\mathcal{I}$ commutes with restriction to open subschemes of $Y$. And if (1) holds then in the proof of Lemma 28.6.1 we showed (2). Thus it suffices to prove that $\mathcal{I}$ is quasi-coherent. Since the property of being quasi-coherent is local we may assume $Y$ is affine. As $f$ is quasi-compact, we can find a finite affine open covering $X = \bigcup_{i = 1, \ldots, n} U_i$. Denote $f'$ the composition $$ X' = \coprod U_i \longrightarrow X \longrightarrow Y. $$ Then $f_*\mathcal{O}_X$ is a subsheaf of $f'_*\mathcal{O}_{X'}$, and hence $\mathcal{I} = \mathop{\rm Ker}(\mathcal{O}_Y \to \mathcal{O}_{X'})$. By Schemes, Lemma 25.24.1 the sheaf $f'_*\mathcal{O}_{X'}$ is quasi-coherent on $Y$. Hence we win. $\square$

    The code snippet corresponding to this tag is a part of the file morphisms.tex and is located in lines 817–834 (see updates for more information).

    \begin{lemma}
    \label{lemma-quasi-compact-scheme-theoretic-image}
    Let $f : X \to Y$ be a morphism of schemes.
    Let $Z \subset Y$ be the scheme theoretic image of $f$.
    If $f$ is quasi-compact then
    \begin{enumerate}
    \item the sheaf of ideals
    $\mathcal{I} = \Ker(\mathcal{O}_Y \to f_*\mathcal{O}_X)$
    is quasi-coherent,
    \item the scheme theoretic image $Z$ is the closed subscheme
    determined by $\mathcal{I}$,
    \item for any open $U \subset Y$ the scheme theoretic image of
    $f|_{f^{-1}(U)} : f^{-1}(U) \to U$ is equal to $Z \cap U$, and
    \item the image $f(X) \subset Z$ is a dense subset of $Z$, in other
    words the morphism $X \to Z$ is dominant
    (see Definition \ref{definition-dominant}).
    \end{enumerate}
    \end{lemma}
    
    \begin{proof}
    Part (4) follows from part (3). To show (3) it suffices
    to prove (1) since the formation of $\mathcal{I}$ commutes with restriction to
    open subschemes of $Y$. And if (1) holds then in the proof of
    Lemma \ref{lemma-scheme-theoretic-image}
    we showed (2). Thus it suffices to prove that $\mathcal{I}$ is quasi-coherent.
    Since the property of being quasi-coherent is
    local we may assume $Y$ is affine. As $f$ is quasi-compact,
    we can find a finite affine open covering
    $X = \bigcup_{i = 1, \ldots, n} U_i$. Denote $f'$ the composition
    $$
    X' = \coprod U_i \longrightarrow X \longrightarrow Y.
    $$
    Then $f_*\mathcal{O}_X$ is a subsheaf of $f'_*\mathcal{O}_{X'}$,
    and hence $\mathcal{I} = \Ker(\mathcal{O}_Y \to \mathcal{O}_{X'})$.
    By Schemes, Lemma \ref{schemes-lemma-push-forward-quasi-coherent}
    the sheaf $f'_*\mathcal{O}_{X'}$ is quasi-coherent on $Y$. Hence we win.
    \end{proof}

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