The Stacks project

Proposition 36.11.2. Let $X$ be a Noetherian scheme. Then the functors

\[ D^-(\textit{Coh}(\mathcal{O}_ X)) \longrightarrow D^-_{\textit{Coh}}(\mathcal{O}_ X) \quad \text{and}\quad D^ b(\textit{Coh}(\mathcal{O}_ X)) \longrightarrow D^ b_{\textit{Coh}}(\mathcal{O}_ X) \]

are equivalences.

Proof. Consider the commutative diagram

\[ \xymatrix{ D^-(\textit{Coh}(\mathcal{O}_ X)) \ar[r] \ar[d] & D^-_{\textit{Coh}}(\mathcal{O}_ X) \ar[d] \\ D^-(\mathit{QCoh}(\mathcal{O}_ X)) \ar[r] & D^-_\mathit{QCoh}(\mathcal{O}_ X) } \]

By Lemma 36.11.1 the left vertical arrow is fully faithful. By Proposition 36.8.3 the bottom arrow is an equivalence. By construction the right vertical arrow is fully faithful. We conclude that the top horizontal arrow is fully faithful. If $K$ is an object of $D^-_{\textit{Coh}}(\mathcal{O}_ X)$ then the object $K'$ of $D^-(\mathit{QCoh}(\mathcal{O}_ X))$ which corresponds to it by Proposition 36.8.3 will have coherent cohomology sheaves. Hence $K'$ is in the essential image of the left vertical arrow by Lemma 36.11.1 and we find that the top horizontal arrow is essentially surjective. This finishes the proof for the bounded above case. The bounded case follows immediately from the bounded above case. $\square$

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Your email address will not be published.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



View Proposition 36.11.2 as pdf