The Stacks project

Lemma 20.19.1. Let $f : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ be a morphism of ringed spaces. Let $y \in Y$. Assume that

  1. $f$ is closed,

  2. $f$ is separated, and

  3. $f^{-1}(y)$ is quasi-compact.

Then for $E$ in $D^+(\mathcal{O}_ X)$ we have $(Rf_*E)_ y = R\Gamma (f^{-1}(y), E|_{f^{-1}(y)})$ in $D^+(\mathcal{O}_{Y, y})$.

Proof. The base change map of Lemma 20.18.1 gives a canonical map $(Rf_*E)_ y \to R\Gamma (f^{-1}(y), E|_{f^{-1}(y)})$. To prove this map is an isomorphism, we represent $E$ by a bounded below complex of injectives $\mathcal{I}^\bullet $. Set $Z = f^{-1}(\{ y\} )$. The assumptions of Lemma 20.17.3 are satisfied, see Topology, Lemma 5.4.2. Hence the restrictions $\mathcal{I}^ n|_ Z$ are acyclic for $\Gamma (Z, -)$. Thus $R\Gamma (Z, E|_ Z)$ is represented by the complex $\Gamma (Z, \mathcal{I}^\bullet |_ Z)$, see Derived Categories, Lemma 13.17.7. In other words, we have to show the map

\[ \mathop{\mathrm{colim}}\nolimits _ V \mathcal{I}^\bullet (f^{-1}(V)) \longrightarrow \Gamma (Z, \mathcal{I}^\bullet |_ Z) \]

is an isomorphism. Using Lemma 20.17.3 we see that it suffices to show that the collection of open neighbourhoods $f^{-1}(V)$ of $Z = f^{-1}(\{ y\} )$ is cofinal in the system of all open neighbourhoods. If $f^{-1}(\{ y\} ) \subset U$ is an open neighbourhood, then as $f$ is closed the set $V = Y \setminus f(X \setminus U)$ is an open neighbourhood of $y$ with $f^{-1}(V) \subset U$. This proves the lemma. $\square$


Comments (0)


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Your email address will not be published.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.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 09V5. Beware of the difference between the letter 'O' and the digit '0'.