4
$\begingroup$

In proposition 2.19. of http://inmabb.criba.edu.ar/revuma/pdf/v48n2/v48n2a05.pdf it was mentioned that a finite dimensional algebra of global dimension 2 is quadratic if and only if it is Koszul.

Question: Is it more general true that a finite dimensional algebra of Gorenstein dimension two is quadratic if and only if it is Koszul?

Here the Gorenstein dimension of an algebra $A$ is the injective dimension of the regular module $A$ (so that the global dimension coincides with the Gorenstein dimension in case the global dimension is finite, thus the question is is more general than the statement in 2.19.).

$\endgroup$
1
  • $\begingroup$ I think the answer is negative: Take a preprojective algebra of Dynkin type A and tensor it with a Koszul algebra of global dimension two. The result is an algebra of Gorenstein dimension two which should not be Koszul. $\endgroup$
    – Mare
    Jul 26, 2022 at 14:30

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.