Lemma 20.22.2. 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 5.23.5). 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 20.19.3. 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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