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The problem is Lemma 13.27.9 of The Stacks Project and the statement is as follows:

Let $\mathcal{A}$ be an abelian category. Let $K$ be an object of $D^b(\mathcal{A})$ such that $\mathrm{Ext}^p_\mathcal{A}(H^i(K),H^j(K))=0$ for all $p\geq2$ and $i>j$. Then $K$ is isomorphic to the direct sum of its cohomologies: $K \cong \bigoplus H^i(K)[−i]$.

In the proof, it says that by using induction we see that $$ \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{A})}(H^ b(K)[-b], (\tau _{\leq b - 1}K)[1]) = \bigoplus \nolimits _{i < b} \mathop{\mathrm{Ext}}\nolimits _\mathcal {A}^{b - i + 1}(H^ b(K), H^ i(K)) $$ but I don't see how it works, even not knowing where to apply the induction, can someone help me on this? Thanks a lot!

The Link to the Lemma 13.27.9

PlatoEinsYu
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  • See this answer/question here for something related: https://math.stackexchange.com/questions/1204016/if-the-cohomology-of-two-objects-in-the-derived-category-are-equal-are-the-obje/1204064#1204064 – Geoff Dec 24 '22 at 20:16
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    @Geoff Thanks! I thought the proofs were irrelevant, I have read that answer again and found that my understanding of this proof wrong. The original text uses two 'induction' words, and now it seems that they are related, we use induction on b+1 and then we can describe the connecting homomorphism with smaller b (or b-a) and get the splitting of b+1, the direct sum equation are from two inductions. – PlatoEinsYu Dec 25 '22 at 03:51

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