Lemma 69.7.1. With $S$, $W$, $G$, $U$, $\chi $ as in Lemma 69.6.5. If $\mathcal{F}$ is a quasi-coherent $\mathcal{O}_ U$-module, then so is $\mathcal{F} \otimes _{\mathbf{Z}} \underline{\mathbf{Z}}(\chi )$.
Proof. The $\mathcal{O}_ U$-module structure is clear. To check that $\mathcal{F} \otimes _{\mathbf{Z}} \underline{\mathbf{Z}}(\chi )$ is quasi-coherent it suffices to check étale locally. Hence the lemma follows as $\underline{\mathbf{Z}}(\chi )$ is finite locally free as a $\underline{\mathbf{Z}}$-module. $\square$
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