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Tag 02KL

Chapter 34: Descent > Section 34.10: Descent of finiteness properties of morphisms

Lemma 34.10.3. Let $$ \xymatrix{ X \ar[rr]_f \ar[rd]_p & & Y \ar[dl]^q \\ & S } $$ be a commutative diagram of morphisms of schemes. Assume that $f$ is surjective, flat and locally of finite presentation and assume that $p$ is locally of finite presentation (resp. locally of finite type). Then $q$ is locally of finite presentation (resp. locally of finite type).

Proof. The problem is local on $S$ and $Y$. Hence we may assume that $S$ and $Y$ are affine. Since $f$ is flat and locally of finite presentation, we see that $f$ is open (Morphisms, Lemma 28.26.9). Hence, since $Y$ is quasi-compact, there exist finitely many affine opens $X_i \subset X$ such that $Y = \bigcup f(X_i)$. Clearly we may replace $X$ by $\coprod X_i$, and hence we may assume $X$ is affine as well. In this case the lemma is equivalent to Lemma 34.10.1 (resp. Lemma 34.10.2) above. $\square$

    The code snippet corresponding to this tag is a part of the file descent.tex and is located in lines 3261–3275 (see updates for more information).

    \begin{lemma}
    \label{lemma-flat-finitely-presented-permanence}
    Let
    $$
    \xymatrix{
    X \ar[rr]_f \ar[rd]_p & &
    Y \ar[dl]^q \\
    & S
    }
    $$
    be a commutative diagram of morphisms of schemes. Assume that $f$ is
    surjective, flat and locally of finite presentation and assume
    that $p$ is locally of finite presentation (resp.\ locally of finite type).
    Then $q$ is locally of finite presentation (resp.\ locally of finite type).
    \end{lemma}
    
    \begin{proof}
    The problem is local on $S$ and $Y$. Hence we may assume that
    $S$ and $Y$ are affine. Since $f$ is flat and locally of finite
    presentation, we see that $f$ is open
    (Morphisms, Lemma \ref{morphisms-lemma-fppf-open}).
    Hence, since $Y$ is quasi-compact, there exist finitely many affine opens
    $X_i \subset X$ such that $Y = \bigcup f(X_i)$.
    Clearly we may replace $X$ by $\coprod X_i$, and hence we
    may assume $X$ is affine as well.
    In this case the lemma is equivalent to
    Lemma \ref{lemma-flat-finitely-presented-permanence-algebra}
    (resp. Lemma \ref{lemma-finite-type-local-source-fppf-algebra})
    above.
    \end{proof}

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