The Stacks project

Lemma 68.9.4. Let $S$ be a scheme. Let $i : Z \to X$ be a closed immersion of algebraic spaces over $S$. The map $Ri_* = i_* : D(Z_{\acute{e}tale}) \to D(X_{\acute{e}tale})$ induces an equivalence $D(Z_{\acute{e}tale}) \to D_{|Z|}(X_{\acute{e}tale})$ with quasi-inverse

\[ i^{-1}|_{D_ Z(X_{\acute{e}tale})} = R\mathcal{H}_ Z|_{D_{|Z|}(X_{\acute{e}tale})} \]

Proof. Recall that $i^{-1}$ and $i_*$ is an adjoint pair of exact functors such that $i^{-1}i_*$ is isomorphic to the identify functor on abelian sheaves. See Properties of Spaces, Lemma 65.19.9 and Morphisms of Spaces, Lemma 66.13.5. Thus $i_* : D(Z_{\acute{e}tale}) \to D_ Z(X_{\acute{e}tale})$ is fully faithful and $i^{-1}$ determines a left inverse. On the other hand, suppose that $K$ is an object of $D_ Z(X_{\acute{e}tale})$ and consider the adjunction map $K \to i_*i^{-1}K$. Using exactness of $i_*$ and $i^{-1}$ this induces the adjunction maps $H^ n(K) \to i_*i^{-1}H^ n(K)$ on cohomology sheaves. Since these cohomology sheaves are supported on $Z$ we see these adjunction maps are isomorphisms and we conclude that $D(Z_{\acute{e}tale}) \to D_ Z(X_{\acute{e}tale})$ is an equivalence.

To finish the proof we have to show that $R\mathcal{H}_ Z(K) = i^{-1}K$ if $K$ is an object of $D_ Z(X_{\acute{e}tale})$. To do this we can use that $K = i_*i^{-1}K$ as we've just proved this is the case. Then we can choose a K-injective representative $\mathcal{I}^\bullet $ for $i^{-1}K$. Since $i_*$ is the right adjoint to the exact functor $i^{-1}$, the complex $i_*\mathcal{I}^\bullet $ is K-injective (Derived Categories, Lemma 13.31.9). We see that $R\mathcal{H}_ Z(K)$ is computed by $\mathcal{H}_ Z(i_*\mathcal{I}^\bullet ) = \mathcal{I}^\bullet $ as desired. $\square$


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Your email address will not be published.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 0AEI. Beware of the difference between the letter 'O' and the digit '0'.