Lemma 73.8.2. Let $S$ be a scheme. Let

$\xymatrix{ X \ar[rr]_ f \ar[rd]_ p & & Y \ar[dl]^ q \\ & B }$

be a commutative diagram of morphisms of algebraic spaces over $S$. 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. Since $\{ X \to Y\}$ is an fppf covering, it induces a surjection of fppf sheaves (Topologies on Spaces, Lemma 72.7.5) and the lemma is a special case of Lemma 73.8.1. On the other hand, an easier argument is to deduce it from the analogue for schemes. Namely, the problem is étale local on $B$ and $Y$ (Morphisms of Spaces, Lemmas 66.23.4 and 66.28.4). Hence we may assume that $B$ and $Y$ are affine schemes. Since $|X| \to |Y|$ is open (Morphisms of Spaces, Lemma 66.30.6), we can choose an affine scheme $U$ and an étale morphism $U \to X$ such that the composition $U \to Y$ is surjective. In this case the result follows from Descent, Lemma 35.14.3. $\square$

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