The Stacks project

Definition 56.15.1. 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 56.13.1 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 $F, F' : D_{perf}(\mathcal{O}_ X) \to D_{perf}(\mathcal{O}_ Y)$.


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