Lemma 13.27.8. 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.**
Choose $a, b$ such that $H^ i(K) = 0$ for $i \not\in [a, b]$. We will prove the lemma by induction on $b - a$. If $b - a \leq 0$, then the result is clear. If $b - a > 0$, then we look at the distinguished triangle of truncations

see Remark 13.12.4. By Lemma 13.4.10 if the last arrow is zero, then $K \cong \tau _{\leq b - 1}K \oplus H^ b(K)[-b]$ and we win by induction. Again using induction we see that

Since $\mathop{\mathrm{Ext}}\nolimits ^ i_\mathcal {A}(B, A) = 0$ for $i \geq 2$ and any pair of objects $A, B$ of $\mathcal{A}$ by our assumption and Lemma 13.27.7 we are done. $\square$

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