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

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. $\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)