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 $n \geq 1$?

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.