# 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 subcategory $$\mathcal{C}$$, 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. (corrected) $$P$$ is a weak generator of the category $$Hot(Proj \,A)$$ if and only if $$P$$ is a classical generator of the category $$Hot^b(proj \,A)$$, that is, the smallest triangulated category containing $$P$$ (which is closed under cones, shifts, isomorphisms and direct summands) is given by the homotopy category of bounded complexes of finitely generated projective $$A$$-modules. This implies that if $$P$$ was a classical generator of $$Hot^b(proj \,A)$$, then $$P$$ would be a weak generator of $$\mathcal{B}$$.

Reference: Stacks-Project, https://stacks.math.columbia.edu/tag/09SR. Please write me if you know the original reference of this fact.

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.

• Are you sure about point 3 in the “background”? The Stacks Project reference you give assumes a triangulated category with arbitrary coproducts, and so applies to the unbounded derived category. But I don’t think that the proof transfers in a straightforward way to the “bounded to the right” derived category. – Jeremy Rickard Jan 18 at 13:20
• This was my mistake. Thank you very much for its correction! – Wayne Jan 21 at 13:29