Given a local commutative (commutative only if needed...) selfinjective finite dimensional algebra $A$ over a field $K$ with enveloping algebra $A^e = A \otimes_K A^{op}$. Then $Ext_{A^{e}}^{1}(A,A) \neq 0$ (here we might need commutativity?). BEcause of $Ext_{A^{e}}^{1}(A,A) \cong \underline{Hom_{A^{e}}}(\Omega_{A^{e}}^{1}(A),A))$ there is a map $\Omega_{A^{e}}^{1}(A) \rightarrow A$ in the stable category representing a nonzero element of $Ext_{A^{e}}^{1}(A,A)$. Question: Given a finite dimensional non-projective module $M$ over $A$, can we tensor a certain such map $\Omega_{A^{e}}^{1}(A) \rightarrow A$ with M to obtain an element $\Omega^{1}(M) \rightarrow M$ in the stable category which represents a nonzero element of $Ext_A^{1}(M,M)$? If the question is too general, we can assume first that $A$ is the group ring of a finite (commutative) p-group.

Given a local commutative (commutative only if needed...) selfinjective (non-semisimple) finite dimensional algebra $A$ over a field $K$ with enveloping algebra $A^e = A \otimes_K A^{op}$. Then $Ext_{A^{e}}^{1}(A,A) \neq 0$ (here we might need commutativity?). BEcause of $Ext_{A^{e}}^{1}(A,A) \cong \underline{Hom_{A^{e}}}(\Omega_{A^{e}}^{1}(A),A))$ there is a map $\Omega_{A^{e}}^{1}(A) \rightarrow A$ in the stable category representing a nonzero element of $Ext_{A^{e}}^{1}(A,A)$. Question: Given a finite dimensional non-projective module $M$ over $A$, can we tensor a certain such map $\Omega_{A^{e}}^{1}(A) \rightarrow A$ with M to obtain an element $\Omega^{1}(M) \rightarrow M$ in the stable category which represents a nonzero element of $Ext_A^{1}(M,M)$? If the question is too general, we can assume first that $A$ is the group ring of a finite (commutative) p-group.

Given a local commutative (commutative only if needed...) selfinjective finite dimensional algebra $A$ over a field $K$ with enveloping algebra $A^e = A \otimes_K A^{op}$. Then $Ext_{A^{e}}^{1}(A,A) \neq 0$ (here we might need commutativity?). BEcause of $Ext_{A^{e}}^{1}(A,A) \cong \underline{Hom_{A^{e}}}(\Omega_{A^{e}}^{1}(A),A))$ there is a map $\Omega_{A^{e}}^{1}(A) \rightarrow A$ in the stable category representing a nonzero element of $Ext_{A^{e}}^{1}(A,A)$. Question: Given a finite dimensional module $M$ over $A$, can we tensor a certain such map $\Omega_{A^{e}}^{1}(A) \rightarrow A$ with M to obtain an element $\Omega^{1}(M) \rightarrow M$ in the stable category which represents a nonzero element of $Ext_A^{1}(M,M)$? If the question is too general, we can assume first that $A$ is the group ring of a finite (commutative) p-group.

Given a local commutative (commutative only if needed...) selfinjective finite dimensional algebra $A$ over a field $K$ with enveloping algebra $A^e = A \otimes_K A^{op}$. Then $Ext_{A^{e}}^{1}(A,A) \neq 0$ (here we might need commutativity?). BEcause of $Ext_{A^{e}}^{1}(A,A) \cong \underline{Hom_{A^{e}}}(\Omega_{A^{e}}^{1}(A),A))$ there is a map $\Omega_{A^{e}}^{1}(A) \rightarrow A$ in the stable category representing a nonzero element of $Ext_{A^{e}}^{1}(A,A)$. Question: Given a finite dimensional non-projective module $M$ over $A$, can we tensor a certain such map $\Omega_{A^{e}}^{1}(A) \rightarrow A$ with M to obtain an element $\Omega^{1}(M) \rightarrow M$ in the stable category which represents a nonzero element of $Ext_A^{1}(M,M)$? If the question is too general, we can assume first that $A$ is the group ring of a finite (commutative) p-group.