Let $P^{\bullet} = (P^i, d^i)_{i\leq 0}$ be an indecomposable object in category of complexes bounded above, where each $P_i$ is a finitely generated projective module over an finite dimensional algebra. Denote by $\tau^{\geq -n} (P^{\bullet})$ the brutal truncation of $P^{\bullet}$, $n \geq 1$, i.e., $$ \tau^{\geq -n} (P^{\bullet}) = \cdots \to 0 \to P^{-n} \to \cdots \to P^{-1} \to P^{0} \to 0 \to \cdots $$

Is it indecomposable $\tau^{\geq -n} (P^{\bullet})$ for all $n \geq 1$? And if in addition we assume that $P^{\bullet}$ is radical (i.e., for each $i$, $\text{Im} \, d^i \subseteq \text{rad} \, P^{i+1}$)?

Thanks!

Edit:

– Suppose $P^{\bullet}$ is unbounded below, i.e., $P^{\bullet} \in \mathcal{C}^{-}(\text{proj}-A) \setminus \mathcal{C}^{b}(\text{proj}-A)$.

– Suppose $P^0 \neq 0$, in order to fix notation.