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Recall that a finite dimensional algebra is called piecewise hereditary in case it is derived equivalent to an abelian hereditary category. In proposition 1.2. of https://link.springer.com/article/10.1007%2FBF01195702 the following was proven:

Let A be a piecewise hereditary algebra with $n$ simple modules and an indecomposable module $X$. Then $pdim(X)+idim(X) \leq n+1$.

It was noted there that the authors do not know whether the bound is attained.

Question: Is it now known whether the bound is attained? In case it is not known, is there an example where it is attained or is the optimal bound $n$ (attained for example at the Kroenecker algebra)?

I do not even know any acyclic algebra where the bound is attained.

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  • $\begingroup$ In your edit, did you mean to write “In case it is known”? $\endgroup$
    – Jeremy Rickard
    Commented Jun 17, 2019 at 22:19
  • $\begingroup$ @JeremyRickard The first question asks for a reference in case the problem was solved. The second question asks for a solution in case there is no existing reference. So people who have no reference but a solution can also answer. (sorry in case this is confusing, but I would expect that the question has been solved) $\endgroup$
    – Mare
    Commented Jun 17, 2019 at 22:22
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    $\begingroup$ For just acyclic, you can take the quiver with three vertices, two arrows from 1 to 2, two arrows from 2 to 3, and three of the four possible zero relations. Then there’s a unique indecomposable representation with dimension vector $(1,1,1)$ which has projective and injective dimensions both equal to two. $\endgroup$
    – Jeremy Rickard
    Commented Jun 17, 2019 at 23:16

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