The Stacks project

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

Lemma 35.14.3. 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 29.25.10). 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 35.14.1 (resp. Lemma 35.14.2) above. $\square$

Comments (2)

Comment #1614 by Kestutis Cesnavicius on

Reference: [EGA IV_4, 17.7.5 (i) and (ii)].

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