The Stacks project

Lemma 13.27.9. Let $\mathcal{A}$ be an abelian category. Let $K$ be an object of $D^ b(\mathcal{A})$ such that $\mathop{\mathrm{Ext}}\nolimits ^ p_\mathcal {A}(H^ i(K), H^ j(K)) = 0$ for all $p \geq 2$ and $i > j$. Then $K$ 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

\[ \tau _{\leq b - 1}K \to K \to H^ b(K)[-b] \to (\tau _{\leq b - 1}K)[1] \]

see Remark 13.12.4. By Lemma 13.4.11 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

\[ \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{A})}(H^ b(K)[-b], (\tau _{\leq b - 1}K)[1]) = \bigoplus \nolimits _{i < b} \mathop{\mathrm{Ext}}\nolimits _\mathcal {A}^{b - i + 1}(H^ b(K), H^ i(K)) \]

By assumption the direct sum is zero and the proof is complete. $\square$

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