The Stacks project

Lemma 20.23.1. Let $X$ be a spectral space. Let $\mathcal{F}$ be an abelian sheaf on $X$. Let $E \subset X$ be a quasi-compact subset. Let $W \subset X$ be the set of points of $X$ which specialize to a point of $E$.

  1. $H^ p(W, \mathcal{F}|_ W) = \mathop{\mathrm{colim}}\nolimits H^ p(U, \mathcal{F})$ where the colimit is over quasi-compact open neighbourhoods of $E$,

  2. $H^ p(W \setminus E, \mathcal{F}|_{W \setminus E}) = \mathop{\mathrm{colim}}\nolimits H^ p(U \setminus E, \mathcal{F}|_{U \setminus E})$ if $E$ is a constructible subset.

Proof. From Topology, Lemma 5.24.7 we see that $W = \mathop{\mathrm{lim}}\nolimits U$ where the limit is over the quasi-compact opens containing $E$. Each $U$ is a spectral space by Topology, Lemma 5.23.4. Thus we may apply Lemma 20.20.2 to conclude that (1) holds. The same proof works for part (2) except we use Topology, Lemma 5.24.8. $\square$


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