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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)

(One might more generally ask whether $\Omega^i(D(A)) \cong \Omega^{i+k}(D(A))$ for some $k \geq 1$, implies $\Omega^{-i}(A) \cong \Omega^{-(i+k)}(A)$)

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 $D(A)=\operatorname{Hom}_K(A,K)$.

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)

(One might more generally ask whether $\Omega^i(D(A)) \cong \Omega^{i+k}(D(A))$ for some $k \geq 1$, implies $\Omega^{-i}(A) \cong \Omega^{-(i+k)}(A)$)

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)

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)

(One might more generally ask whether $\Omega^i(D(A)) \cong \Omega^{i+k}(D(A))$ for some $k \geq 1$, implies $\Omega^{-i}(A) \cong \Omega^{-(i+k)}(A)$)

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 $\Omega^{-i}(A) \cong \Omega^{-(i+1)}(A)$?

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)