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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