Lemma 13.10.6. Let \mathcal{A}, \mathcal{B} be additive categories. Let F : \mathcal{A} \to \mathcal{B} be an additive functor. The induced functors
\begin{matrix} F : K(\mathcal{A}) \longrightarrow K(\mathcal{B})
\\ F : K^{+}(\mathcal{A}) \longrightarrow K^{+}(\mathcal{B})
\\ F : K^{-}(\mathcal{A}) \longrightarrow K^{-}(\mathcal{B})
\\ F : K^ b(\mathcal{A}) \longrightarrow K^ b(\mathcal{B})
\end{matrix}
are exact functors of triangulated categories.
Proof.
Suppose A^\bullet \to B^\bullet \to C^\bullet is a termwise split sequence of complexes of \mathcal{A} with splittings (s^ n, \pi ^ n) and associated morphism \delta : C^\bullet \to A^\bullet [1], see Definition 13.9.9. Then F(A^\bullet ) \to F(B^\bullet ) \to F(C^\bullet ) is a termwise split sequence of complexes with splittings (F(s^ n), F(\pi ^ n)) and associated morphism F(\delta ) : F(C^\bullet ) \to F(A^\bullet )[1]. Thus F transforms distinguished triangles into distinguished triangles.
\square
Comments (0)