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.


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.