Given a finite dimensional algebra $A$ over a field $K$ with $\Omega^i(D(A)) \cong \Omega^{i+1}(D(A))$ for some $i \geq 1$, where $D(A)=Hom_K(A,K)$. Do we then also have $\Omega^{-i}(A) \cong \Omega^{-(i+1)}(A)$? In case the answer is yes, do we even have that $\Omega^{-i}(A)$ and $\Omega^i(D(A))$ have the same vector space dimension? (Note that a positive answer would prove the Gorenstein symmetry conjecture) Login To add answer/comment