# Weak generators of the right-bounded derived category of a finite-dimensional algebra

The setup:

• Let $$A$$ be a finite-dimensional $$k$$-algebra over some field $$k$$.
• Let $$\mathcal{B} = Hot^-(Proj A)$$ denote the homotopy category of cochain complexes of (possibly infinitely generated) projective right $$A$$-modules which are bounded from the right. This category can be identified with the right-bounded derived category $$D^-(Mod A)$$ of $$A$$-modules.
• Let $$\mathcal{C} = Hot^-(proj A)$$ denote the full subcategory of $$\mathcal{B}$$ given by right-bounded complexes of finitely generated projective $$A$$-modules. In different terms, this category corresponds to the right-bounded derived category $$D^-(mod A)$$ of finitely generated $$A$$-modules.
• Let $$P$$ be a perfect object of $$\mathcal{B}$$, that is, a bounded complex of finitely generated projective right $$A$$-modules. Assume also that $$P$$ is a weak generator of the small category $$\mathcal{C}$$, so for any object $$X \in \mathcal{C}$$ there is some integer $$m$$ and some non-zero morphism $$P \to X[m]$$ in $$\mathcal{C}$$.

My question:

Is $$P$$ already a weak generator of the big category $$\mathcal{B}$$?

Some background:

1. By a result of Jeremy Rickard, the answer is affirmative if $$P$$ is a partial tilting complex, that is, if $$Hom_{\mathcal{B}}(P,P[n])=0$$ for any non-zero integer $$n$$.

Reference: Proposition 5.4 in Rickard, Jeremy, Morita theory for derived categories, J. Lond. Math. Soc., II. Ser. 39, No. 3, 436-456 (1989). ZBL0642.16034.

1. By a result of Bruno J. Müller, any projective $$A$$-module is a (possibly infinite) direct sum of finitely generated projective $$A$$-modules.

Reference: Müller, Bruno J., On semi-perfect rings, Mathematical Report No. 19, Vol. 1. Hamilton, Ont.: McMaster University, Department of Mathematics. 11 p. (1969). ZBL0226.16026.

1. At least, $$P$$ is a weak generator of the homotopy category of complexes of projective $$A$$-modules which have finitely generated cohomology at each degree.

Any comments and any input will be very appreciated.

• Given the rather advanced nature of this question, I think it would receive more attention on MathOverflow. If you agree, you can click flag>in need of moderator intervention and request that a moderator migrate the question to MathOverflow. – Mike Pierce Jan 14 at 19:41
• Thank you for the advice. I have posted a slightly extended version of the question on MathOverflow. – Wayne Jan 16 at 18:50