The Stacks project

Definition 57.13.1. Let $k$ be a field. Let $X$, $Y$ be finite type schemes over $k$. Recall that $D^ b_{\textit{Coh}}(\mathcal{O}_ X) = D^ b(\textit{Coh}(\mathcal{O}_ X))$ by Derived Categories of Schemes, Proposition 36.11.2. We say two $k$-linear exact functors

\[ F, F' : D^ b_{\textit{Coh}}(\mathcal{O}_ X) = D^ b(\textit{Coh}(\mathcal{O}_ X)) \longrightarrow D^ b_{\textit{Coh}}(\mathcal{O}_ Y) \]

are siblings, or we say $F'$ is a sibling of $F$ if $F$ and $F'$ are siblings in the sense of Definition 57.11.1 with abelian category being $\textit{Coh}(\mathcal{O}_ X)$. If $X$ is regular then $D_{perf}(\mathcal{O}_ X) = D^ b_{\textit{Coh}}(\mathcal{O}_ X)$ by Derived Categories of Schemes, Lemma 36.11.6 and we use the same terminology for $k$-linear exact functors $F, F' : D_{perf}(\mathcal{O}_ X) \to D_{perf}(\mathcal{O}_ Y)$.

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Your email address will not be published.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