Lemma 33.38.2 (09N9)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 33: Varieties
Section 33.38: One dimensional Noetherian schemes
Lemma 33.38.2 (cite)

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Lemma 33.38.2. Let $X$ be an affine scheme all of whose local rings are Noetherian of dimension $\leq 1$. Then any quasi-compact open $U \subset X$ is affine.

Proof. Denote $j : U \to X$ the inclusion morphism. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ U$-module. By Lemma 33.38.1 the higher direct images $R^ pj_*\mathcal{F}$ are zero. The $\mathcal{O}_ X$-module $j_*\mathcal{F}$ is quasi-coherent (Schemes, Lemma 26.24.1). Hence it has vanishing higher cohomology groups by Cohomology of Schemes, Lemma 30.2.2. By the Leray spectral sequence Cohomology, Lemma 20.13.6 we have $H^ p(U, \mathcal{F}) = 0$ for all $p > 0$. Thus $U$ is affine, for example by Cohomology of Schemes, Lemma 30.3.1. $\square$

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