Lemma 51.12.2 (0BLT)—The Stacks project

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Part 3: Topics in Scheme Theory
Chapter 51: Local Cohomology
Section 51.12: Finiteness of pushforwards, II
Lemma 51.12.2 (cite)

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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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