Lemma 85.28.2. Let $f : V \to U$ be a cartesian morphism of simplicial schemes. Assume the morphisms $d^ n_ j : U_ n \to U_{n - 1}$ are flat and the morphisms $V_ n \to U_ n$ are quasi-compact and quasi-separated. For a quasi-coherent module $\mathcal{G}$ on $V_{Zar}$ the pushforward $f_*\mathcal{G}$ is a quasi-coherent module on $U_{Zar}$.

Proof. If $\mathcal{F} = f_* \mathcal{G}$, then $\mathcal{F}_ n = f_{n , *}\mathcal{G}_ n$ by Lemma 85.2.4. The maps $\mathcal{F}(\varphi )$ are defined using the base change maps, see Cohomology, Section 20.17. The sheaves $\mathcal{F}_ n$ are quasi-coherent by Schemes, Lemma 26.24.1 and the fact that $\mathcal{G}_ n$ is quasi-coherent by Lemma 85.12.10. The base change maps along the degeneracies $d^ n_ j$ are isomorphisms by Cohomology of Schemes, Lemma 30.5.2 and the fact that $\mathcal{G}$ is cartesian by Lemma 85.12.10. Hence $\mathcal{F}$ is cartesian by Lemma 85.12.2. Thus $\mathcal{F}$ is quasi-coherent by Lemma 85.12.10. $\square$

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