Given a finite dimensional connected algebra $A$ with $Ext^{1}(M,M) \neq 0$ for any non-projective and non-injective module $M$.


1. Is $A$ selfinjective?
2. Is $A$ local?



Algebra with all modules non-rigid
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 $$\dim \operatorname{Ext}^1(M,M)=\dim \operatorname{Ext}^1(S(1)^{n\_1},S(2)^{n\_2})=n\_1n\_2\neq 0.$$ 


