Given a finite dimensional connected algebra $A$ with $Ext^{1}(M,M) \neq 0$ for any non-projective and non-injective module $M$.
Is $A$ selfinjective?
Is $A$ local?
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2$\begingroup$ You can probably guess my pedantic question about (2) ... :-) $\endgroup$– Jeremy RickardCommented Jul 22, 2017 at 15:15
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2$\begingroup$ I added "connected". $\endgroup$– MareCommented Jul 22, 2017 at 15:23
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1 Answer
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No, the path algebra of an $A_2$-quiver provides a counterexample. Let $M$ be a (finite dimensional) module. Then $M=S(1)^{n_1}\oplus S(2)^{n_2}\oplus P(1)^{n_3}$ where $S(1)=I(1)$ is injective non-projective and $S(2)=P(2)$ is projective non-injective. That $M$ is non-projective is thus equivalent to $n_1\neq 0$, the fact that $M$ is non-injective means that $n_2\neq 0$. But then $$\dim \operatorname{Ext}^1(M,M)=\dim \operatorname{Ext}^1(S(1)^{n_1},S(2)^{n_2})=n_1n_2\neq 0.$$
answered Jul 24, 2017 at 8:10
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$\begingroup$ Thanks, I actually wanted to ask this question with M indecomposable but forgot this condition. I make a new thread. Here any indecomposable is projective or injective. $\endgroup$– MareCommented Jul 24, 2017 at 8:36