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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.).

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  • $\begingroup$ 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
    Commented Jul 26, 2022 at 14:30

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