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Tag 03GI

Chapter 25: Schemes > Section 25.21: Separation axioms

Lemma 25.21.15. Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of schemes. If $g \circ f$ is quasi-compact and $g$ is quasi-separated then $f$ is quasi-compact.

Proof. This is true because $f$ equals the composition $(1, f) : X \to X \times_Z Y \to Y$. The first map is quasi-compact by Lemma 25.21.12 because it is a section of the quasi-separated morphism $X \times_Z Y \to X$ (a base change of $g$, see Lemma 25.21.13). The second map is quasi-compact as it is the base change of $f$, see Lemma 25.19.3. And compositions of quasi-compact morphisms are quasi-compact, see Lemma 25.19.4. $\square$

    The code snippet corresponding to this tag is a part of the file schemes.tex and is located in lines 4194–4199 (see updates for more information).

    \begin{lemma}
    \label{lemma-quasi-compact-permanence}
    Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of schemes.
    If $g \circ f$ is quasi-compact and $g$ is quasi-separated
    then $f$ is quasi-compact.
    \end{lemma}
    
    \begin{proof}
    This is true because $f$ equals the composition
    $(1, f) : X \to X \times_Z Y \to Y$. The first map
    is quasi-compact by Lemma \ref{lemma-section-immersion}
    because it is a section of the quasi-separated morphism $X \times_Z Y \to X$
    (a base change of $g$, see Lemma \ref{lemma-separated-permanence}).
    The second map is quasi-compact as it
    is the base change of $f$, see
    Lemma \ref{lemma-quasi-compact-preserved-base-change}.
    And compositions of quasi-compact
    morphisms are quasi-compact, see Lemma \ref{lemma-composition-quasi-compact}.
    \end{proof}

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