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Tag 0462

Chapter 58: Morphisms of Algebraic Spaces > Section 58.23: Morphisms of finite type

Lemma 58.23.6. Let $S$ be a scheme. Let $f : X \to Y$, $g : Y \to Z$ be morphisms of algebraic spaces over $S$. If $g \circ f : X \to Z$ is locally of finite type, then $f : X \to Y$ is locally of finite type.

Proof. We can find a diagram $$ \xymatrix{ U \ar[r] \ar[d] & V \ar[r] \ar[d] & W \ar[d] \\ X \ar[r] & Y \ar[r] & Z } $$ where $U$, $V$, $W$ are schemes, the vertical arrows are étale and surjective, see Spaces, Lemma 56.11.6. At this point we can use Lemma 58.23.4 and Morphisms, Lemma 28.14.8 to conclude. $\square$

    The code snippet corresponding to this tag is a part of the file spaces-morphisms.tex and is located in lines 4463–4469 (see updates for more information).

    \begin{lemma}
    \label{lemma-permanence-finite-type}
    Let $S$ be a scheme.
    Let $f : X \to Y$, $g : Y \to Z$ be morphisms of algebraic spaces over $S$.
    If $g \circ f : X \to Z$ is locally of finite type, then $f : X \to Y$
    is locally of finite type.
    \end{lemma}
    
    \begin{proof}
    We can find a diagram
    $$
    \xymatrix{
    U \ar[r] \ar[d] & V \ar[r] \ar[d] & W \ar[d] \\
    X \ar[r] & Y \ar[r] & Z
    }
    $$
    where $U$, $V$, $W$ are schemes, the vertical arrows are \'etale and surjective,
    see
    Spaces, Lemma \ref{spaces-lemma-lift-morphism-presentations}.
    At this point we can use
    Lemma \ref{lemma-finite-type-local}
    and
    Morphisms, Lemma \ref{morphisms-lemma-permanence-finite-type}
    to conclude.
    \end{proof}

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