If an object of an abelian category is split in the derived category, then must it be split?

Question

Let $\mathcal{A}$ be an abelian category, and $A\in\mathcal{A}$ an object. Let $D(\mathcal{A})$ be its derived category, and let $A'$ denote the image of $A$ in $D(\mathcal{A})$. If $A' = B'\oplus C'$ in the derived category (ie, is split in $D(\mathcal{A})$), then must $A$ be split in $\mathcal{A}$?

Answer 1

Firstly, the way we view $A$ inside $D(\mathcal{A})$ is of course via the complex "$A[0]$" concentrated in degree 0: $$\cdots\rightarrow 0\rightarrow A\rightarrow 0\rightarrow\cdots$$ If it were split in the derived category, then $A[0]$ must be quasi-isomorphic to $B^\bullet\oplus C^\bullet$, where both $B^\bullet,C^\bullet$ are not quasi-isomorphic to 0.

Since $H^i(B^\bullet\oplus C^\bullet) = H^i(B^\bullet)\oplus H^i(C^\bullet)$, we must have $H^0(B^\bullet)\oplus H^0(C^\bullet)\cong A$, $H^i(B^\bullet)\oplus H^i(C^\bullet)\cong 0$ for all $i\ne 0$, and hence $H^i(B^\bullet),H^i(C^\bullet)$ are each individually $\cong 0$ for all $i\ne 0$.

Thus, if $A$ is indecomposable in $\mathcal{A}$, then we may assume $H^0(B^\bullet) = A$, and $H^0(C^\bullet) = 0$, which shows that $C^\bullet$ is quasi-isomorphic to 0, so $A[0]$ cannot be split in $D(\mathcal{A})$.