Lemma 36.6.3. Let $X$ be a scheme. Let $T \subset X$ be a closed subset such that $X \setminus T$ is a retrocompact open of $X$. Then for a family of objects $E_ i$, $i \in I$ of $D_\mathit{QCoh}(\mathcal{O}_ X)$ we have $R\mathcal{H}_ T(\bigoplus E_ i) = \bigoplus R\mathcal{H}_ T(E_ i)$.
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 $D(\mathcal{O}_ X)$ for any $E$ in $D(\mathcal{O}_ X)$. The functor $E \mapsto Rj_*(E|_ U)$ commutes with direct sums on $D_\mathit{QCoh}(\mathcal{O}_ X)$ by Lemma 36.4.5. It follows that the same is true for the functor $i_* \circ R\mathcal{H}_ T$ (details omitted). Since $i_* : D(i^{-1}\mathcal{O}_ X) \to D_ T(\mathcal{O}_ X)$ is an equivalence (Cohomology, Lemma 20.34.2) we conclude. $\square$
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Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)
There are also: