## 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 28.25.9). 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 3305–3322 (see updates for more information).

```
\begin{lemma}
\label{lemma-flat-finitely-presented-permanence}
\begin{reference}
\cite[IV, 17.7.5 (i) and (ii)]{EGA}.
\end{reference}
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}
```

## References

[EGA, IV, 17.7.5 (i) and (ii)].

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