Lemma 13.28.4. Let H : \mathcal{D} \to \mathcal{A} be a homological functor from a triangulated category to an abelian category. Assume that for any X in \mathcal{D} only a finite number of the objects H(X[i]) are nonzero in \mathcal{A}. Then H induces a group homomorphism K_0(\mathcal{D}) \to K_0(\mathcal{A}) sending [X] to \sum (-1)^ i[H(X[i])].
Proof. Omitted. \square
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