The Stacks project

Lemma 13.11.1. Let $\mathcal{A}$ be an abelian category. The functor

\[ H^0 : K(\mathcal{A}) \longrightarrow \mathcal{A} \]

is homological.

Proof. Because $H^0$ is a functor, and by our definition of distinguished triangles it suffices to prove that given a termwise split short exact sequence of complexes $0 \to A^\bullet \to B^\bullet \to C^\bullet \to 0$ the sequence $H^0(A^\bullet ) \to H^0(B^\bullet ) \to H^0(C^\bullet )$ is exact. This follows from Homology, Lemma 12.13.12. $\square$

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