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Let $A$ be a quiver algebra with global dimension at most two (or more generally finite global dimension) and $A^e=A^{op} \otimes_K A$ be its enveloping algebra.

Guess:Is $A$ hereditary if and only if the socle of the regular module as a bimodule $soc_{A^e}(A)$ is projective?

I think I can prove one direction in case $A=kQ$ is a path algebra, which would prove this direction for example for general finite dimensional algebra over an algebraically closed field.

Namely because there are no relations, we should have that $soc_{A^e}(A)$ is the direct sum of all simple projective modules of the form $p A^e$ for $p$ a maximal path in $A$ (meaning $p \neq 0$ and $xp=0=px$ for all arrows x).

At least for quiver algebras having the property that between any two points there is at most one path the bimodule $soc_{A^e}(A)$ can be described as the module of maximal paths in that algebra. Im not sure if this is a good general description for algebras with more complicated relations...

I tested the guess mostly for Nakayama algebras and there the guess has a positive answer for all algebras with at most 8 simples modules. But I have no good idea for proving the other direction.

Question :The projective dimension of $soc_{A^e}(A)$ is bounded by one iff $A$ is a tilted (or quasi-tilted?) algebra? (I think it is very surprising that this holds even for Nakayama algebras with at most 8 simples)

edit: Jan Geuenich found an example of an algebra with infinite global dimension that gives negative answers to the posed question/guess. I restrict therefore to algebras of finite global dimension since I think there might still be some positive results with possibly stronger assumptions.

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