3
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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.

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  • $\begingroup$ 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. $\endgroup$ – Jeremy Rickard Jan 18 at 13:20
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    $\begingroup$ This was my mistake. Thank you very much for its correction! $\endgroup$ – Wayne Jan 21 at 13:29

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