The Stacks project

Lemma 33.38.1. Let $X$ be a scheme all of whose local rings are Noetherian of dimension $\leq 1$. Let $U \subset X$ be a retrocompact open. Denote $j : U \to X$ the inclusion morphism. Then $R^ pj_*\mathcal{F} = 0$, $p > 0$ for every quasi-coherent $\mathcal{O}_ U$-module $\mathcal{F}$.

Proof. We may check the vanishing of $R^ pj_*\mathcal{F}$ at stalks. Formation of $R^ qj_*$ commutes with flat base change, see Cohomology of Schemes, Lemma 30.5.2. Thus we may assume that $X$ is the spectrum of a Noetherian local ring of dimension $\leq 1$. In this case $X$ has a closed point $x$ and finitely many other points $x_1, \ldots , x_ n$ which specialize to $x$ but not each other (see Algebra, Lemma 10.31.6). If $x \in U$, then $U = X$ and the result is clear. If not, then $U = \{ x_1, \ldots , x_ r\} $ for some $r$ after possibly renumbering the points. Then $U$ is affine (Schemes, Lemma 26.11.8). Thus the result follows from Cohomology of Schemes, Lemma 30.2.3. $\square$


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