Remark 57.13.3. If $F, F' : D^ b_{\textit{Coh}}(\mathcal{O}_ X) \to \mathcal{D}$ are siblings, $F$ is fully faithful, and $X$ is reduced and projective over $k$ then $F \cong F'$; this follows from Proposition 57.11.6 via the argument given in the proof of Theorem 57.14.3. However, in general we do not know whether siblings are isomorphic. Even in the situation of Lemma 57.13.2 it seems difficult to prove that the siblings $F$ and $F'$ are isomorphic functors. If $X$ is smooth and proper over $k$ and $F$ is fully faithful, then $F \cong F'$ as is shown in [Noah]. If you have a proof or a counter example in more general situations, please email stacks.project@gmail.com.

## 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Your email address will not be published.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 0G01. Beware of the difference between the letter 'O' and the digit '0'.