The Stacks project

Lemma 36.6.2. Let $X$ be a scheme. Let $T \subset X$ be a closed subset such that $X \setminus T$ is a retrocompact open of $X$. Let $i : T \to X$ be the inclusion.

  1. For $E$ in $D_\mathit{QCoh}(\mathcal{O}_ X)$ we have $i_*R\mathcal{H}_ T(E)$ in $D_{\mathit{QCoh}, T}(\mathcal{O}_ X)$.

  2. The functor $i_* \circ R\mathcal{H}_ T : D_\mathit{QCoh}(\mathcal{O}_ X) \to D_{\mathit{QCoh}, T}(\mathcal{O}_ X)$ is right adjoint to the inclusion functor $D_{\mathit{QCoh}, T}(\mathcal{O}_ X) \to D_\mathit{QCoh}(\mathcal{O}_ X)$.

Proof. Set $U = X \setminus T$ and denote $j : U \to X$ the inclusion. By Cohomology, Lemma 20.34.6 there is a distinguished triangle

\[ i_*R\mathcal{H}_ T(E) \to E \to Rj_*(E|_ U) \to i_*R\mathcal{H}_ Z(E)[1] \]

in $D(\mathcal{O}_ X)$. By Lemma 36.4.1 the complex $Rj_*(E|_ U)$ has quasi-coherent cohomology sheaves (this is where we use that $U$ is retrocompact in $X$). Thus we see that (1) is true. Part (2) follows from this and the adjointness of functors in Cohomology, Lemma 20.34.2. $\square$

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Your email address will not be published.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 0G7G. Beware of the difference between the letter 'O' and the digit '0'.