Lemma 74.8.2. 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 73.7.5) and the lemma is a special case of Lemma 74.8.1. 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 67.23.4 and 67.28.4). Hence we may assume that B and Y are affine schemes. Since |X| \to |Y| is open (Morphisms of Spaces, Lemma 67.30.6), 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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