The Stacks project

Lemma 22.27.17. Let $R$ be a ring. Let $F : \mathcal{A} \to \mathcal{B}$ be a functor between differential graded categories over $R$ satisfying axioms (A), (B), and (C) such that $F(x[1]) = F(x)[1]$. Then $F$ induces an exact functor $K(\mathcal{A}) \to K(\mathcal{B})$ of triangulated categories.

Proof. Namely, if $x \to y \to z$ is an admissible short exact sequence in $\text{Comp}(\mathcal{A})$, then $F(x) \to F(y) \to F(z)$ is an admissible short exact sequence in $\text{Comp}(\mathcal{B})$. Moreover, the “boundary” morphism $\delta = \pi \text{d}(s) : z \to x[1]$ constructed in Lemma 22.27.1 produces the morphism $F(\delta ) : F(z) \to F(x[1]) = F(x)[1]$ which is equal to the boundary map $F(\pi ) \text{d}(F(s))$ for the admissible short exact sequence $F(x) \to F(y) \to F(z)$. $\square$

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