The Stacks project

Lemma 51.12.2. Let $X$ be a locally Noetherian scheme. Let $j : U \to X$ be the inclusion of an open subscheme with complement $Z$. Let $n \geq 0$ be an integer. If $R^ pj_*\mathcal{O}_ U$ is coherent for $0 \leq p < n$, then the same is true for $R^ pj_*\mathcal{F}$, $0 \leq p < n$ for any finite locally free $\mathcal{O}_ U$-module $\mathcal{F}$.

Proof. The question is local on $X$, hence we may assume $X$ is affine. Say $X = \mathop{\mathrm{Spec}}(A)$ and $Z = V(I)$. Via Lemma 51.8.2 our lemma follows from Lemma 51.7.4. $\square$

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