Whether a partial tilting complex has a complement

Question

I want to know whether a partial tilting complex has a complement。if the answer is obvious?or to what kind of algebra this is true。

Answer 1

There's a simple counterexample in Section 8 of

Rickard, Jeremy, Morita theory for derived categories, J. Lond. Math. Soc., II. Ser. 39, No. 3, 436-456 (1989). ZBL0642.16034.

Let kQ be the path algebra of the quiver $$1\longrightarrow 2\longrightarrow 3,$$ and let $T:= \dots\to0\to P_2\to P_1\to0\to\dots$ be a minimal projective resolution of the simple at vertex $1$. Then $P_3\oplus T[n]$ is a partial tilting complex for any $n$, but for most $n$ it has no complement.