The Stacks project

Lemma 20.23.2. Let $f : X \to Y$ be a spectral map of spectral spaces. Let $y \in Y$. Let $E \subset Y$ be the set of points specializing to $y$. Let $\mathcal{F}$ be an abelian sheaf on $X$. Then $(R^ pf_*\mathcal{F})_ y = H^ p(f^{-1}(E), \mathcal{F}|_{f^{-1}(E)})$.

Proof. Observe that $E = \bigcap V$ where $V$ runs over the quasi-compact open neighbourhoods of $y$ in $Y$. Hence $f^{-1}(E) = \bigcap f^{-1}(V)$. This implies that $f^{-1}(E) = \mathop{\mathrm{lim}}\nolimits f^{-1}(V)$ as topological spaces. Since $f$ is spectral, each $f^{-1}(V)$ is a spectral space too (Topology, Lemma 5.23.4). We conclude that $f^{-1}(E)$ is a spectral space and that

\[ H^ p(f^{-1}(E), \mathcal{F}|_{f^{-1}(E)}) = \mathop{\mathrm{colim}}\nolimits H^ p(f^{-1}(V), \mathcal{F}) \]

by Lemma 20.20.2. On the other hand, the stalk of $R^ pf_*\mathcal{F}$ at $y$ is given by the colimit on the right. $\square$


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