Lemma 13.27.10. Let $\mathcal{A}$ be an abelian category. Assume $\mathop{\mathrm{Ext}}\nolimits ^2_\mathcal {A}(B, A) = 0$ for any pair of objects $A$, $B$ of $\mathcal{A}$. Then any object $K$ of $D^ b(\mathcal{A})$ is isomorphic to the direct sum of its cohomologies: $K \cong \bigoplus H^ i(K)[-i]$.

**Proof.**
The assumption implies that $\mathop{\mathrm{Ext}}\nolimits ^ i_\mathcal {A}(B, A) = 0$ for $i \geq 2$ and any pair of objects $A, B$ of $\mathcal{A}$ by Lemma 13.27.8. Hence this lemma is a special case of Lemma 13.27.9.
$\square$

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