The Stacks project

Lemma 22.35.2. The functor of Lemma 22.35.1 defines an exact functor of triangulated categories $K(\text{Mod}_{(E, \text{d})}) \to K(\mathcal{O})$.

Proof. The functor induces a functor between homotopy categories by Lemma 22.26.5. We have to show that $- \otimes _ E K^\bullet $ transforms distinguished triangles into distinguished triangles. Suppose that $0 \to K \to L \to M \to 0$ is an admissible short exact sequence of differential graded $E$-modules. Let $s : M \to L$ be a graded $E$-module homomorphism which is left inverse to $L \to M$. Then $s$ defines a map $M \otimes _ E K^\bullet \to L \otimes _ E K^\bullet $ of graded $\mathcal{O}$-modules (i.e., respecting $\mathcal{O}$-module structure and grading, but not differentials) which is left inverse to $L \otimes _ E K^\bullet \to M \otimes _ E K^\bullet $. Thus we see that

\[ 0 \to K \otimes _ E K^\bullet \to L \otimes _ E K^\bullet \to M \otimes _ E K^\bullet \to 0 \]

is a termwise split short exact sequences of complexes, i.e., a defines a distinguished triangle in $K(\mathcal{O})$. $\square$

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