The Stacks project

Lemma 36.11.6. Let $X$ be a Noetherian scheme. Let $E$ in $D(\mathcal{O}_ X)$ be perfect. Then

  1. $E$ is in $D^ b_{\textit{Coh}}(\mathcal{O}_ X)$,

  2. if $L$ is in $D_{\textit{Coh}}(\mathcal{O}_ X)$ then $E \otimes _{\mathcal{O}_ X}^\mathbf {L} L$ and $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(E, L)$ are in $D_{\textit{Coh}}(\mathcal{O}_ X)$,

  3. if $L$ is in $D^ b_{\textit{Coh}}(\mathcal{O}_ X)$ then $E \otimes _{\mathcal{O}_ X}^\mathbf {L} L$ and $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(E, L)$ are in $D^ b_{\textit{Coh}}(\mathcal{O}_ X)$,

  4. if $L$ is in $D^+_{\textit{Coh}}(\mathcal{O}_ X)$ then $E \otimes _{\mathcal{O}_ X}^\mathbf {L} L$ and $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(E, L)$ are in $D^+_{\textit{Coh}}(\mathcal{O}_ X)$,

  5. if $L$ is in $D^-_{\textit{Coh}}(\mathcal{O}_ X)$ then $E \otimes _{\mathcal{O}_ X}^\mathbf {L} L$ and $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(E, L)$ are in $D^-_{\textit{Coh}}(\mathcal{O}_ X)$.

Proof. Since $X$ is quasi-compact, each of these statements can be checked over the members of any open covering of $X$. Thus we may assume $E$ is represented by a bounded complex $\mathcal{E}^\bullet $ of finite free modules, see Cohomology, Lemma 20.46.3. In this case each of the statements is clear as both $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(E, L)$ and $E \otimes _{\mathcal{O}_ X}^\mathbf {L} L$ can be computed on the level of complexes using $\mathcal{E}^\bullet $, see Cohomology, Lemmas 20.43.9 and 20.26.9. Some details omitted. $\square$


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Your email address will not be published.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