The Stacks project

Lemma 84.28.1. Let $f : V \to U$ be a morphism of simplicial schemes. Given a quasi-coherent module $\mathcal{F}$ on $U_{Zar}$ the pullback $f^*\mathcal{F}$ is a quasi-coherent module on $V_{Zar}$.

Proof. Recall that $\mathcal{F}$ is cartesian with $\mathcal{F}_ n$ quasi-coherent, see Lemma 84.12.10. By Lemma 84.2.4 we see that $(f^*\mathcal{F})_ n = f_ n^*\mathcal{F}_ n$ (some details omitted). Hence $(f^*\mathcal{F})_ n$ is quasi-coherent. The same fact and the cartesian property for $\mathcal{F}$ imply the cartesian property for $f^*\mathcal{F}$. Thus $\mathcal{F}$ is quasi-coherent by Lemma 84.12.10 again. $\square$

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