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.
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$\begingroup$ Are you expecting such extension group to be nonvanishing always? Because it isn't. $\endgroup$– PedroCommented Sep 21, 2016 at 16:17
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$\begingroup$ I added "nonprojective" to M. Id be very surprised if you have an easy example of a nonprojective module over such algebras with Ext^1 vanishing. $\endgroup$– MareCommented Sep 21, 2016 at 16:19
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1$\begingroup$ Maybe Peter was thinking in $Ext_{A^{e}}^{1}(A,A)$? (take $A=K$). $\endgroup$– A.GCommented Sep 21, 2016 at 17:52
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$\begingroup$ ok, i forgot the assumption non-semisimple of course, but in that case all modules are projective anyway. $\endgroup$– MareCommented Sep 21, 2016 at 17:55
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