2
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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. $\endgroup$ – Mike Pierce Jan 14 at 19:41
  • $\begingroup$ Thank you for the advice. I have posted a slightly extended version of the question on MathOverflow. $\endgroup$ – Wayne Jan 16 at 18:50

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.