Lemma 15.83.11. Let $R$ be a ring. Let $K_ j \in D(R)$, $j = 1, 2, 3$ with $H^ i(K_ j) = 0$ for $i \not\in \{ -1, 0\} $. Let $\varphi : K_1 \to K_2$ and $\psi : K_2 \to K_3$ be maps in $D(R)$. If $H^0(\varphi ) = 0$ and $H^{-1}(\psi ) = 0$, then $\varphi \circ \psi = 0$.

**Proof.**
Apply Derived Categories, Lemma 13.12.5 to see that $\varphi \circ \psi $ factors through $\tau _{\leq -2}K_2 = 0$.
$\square$

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