Lemma 37.34.2. Let f : X \to S be a morphism of affine schemes, which is of finite presentation. Let \mathcal{F} be a quasi-coherent \mathcal{O}_ X-module of finite presentation. Then there exists a diagram as in Lemma 37.34.1 such that there exists a coherent \mathcal{O}_{X_0}-module \mathcal{F}_0 with g^*\mathcal{F}_0 = \mathcal{F}.
Proof. Write S = \mathop{\mathrm{Spec}}(A), X = \mathop{\mathrm{Spec}}(B), and \mathcal{F} = \widetilde{M}. As f is of finite presentation we see that B is of finite presentation as an A-algebra, see Morphisms, Lemma 29.21.2. As \mathcal{F} is of finite presentation over \mathcal{O}_ X we see that M is of finite presentation as a B-module, see Properties, Lemma 28.16.2. Thus the lemma follows from Algebra, Lemma 10.127.18. \square
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