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Given a finite dimensional algebra $A$ over an infinite field and two simple modules $S,T$.

Question 1: Is there a useful (homological/computational) crtierion to decide when there are infinitely many length 2 indecomposable modules $M$ that are appear as an extension of $S,T$? Can one read this off somehow by knowing $Ext_A^1(T,S)$ as a vector space, possibly with some extra structure?

So this means we have an exact sequence $0 \rightarrow S \rightarrow M \rightarrow T \rightarrow 0$ or equivalently $M$ is indecomposable of length 2 with socle $S$ and top $T$.

Question 2: Which quiver algebras have only finitely many two-dimensional indecomposable modules?

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    $\begingroup$ If you just ask for bound quiver algebras, the answer is $\operatorname{dim}_k \operatorname{Ext}^1(T,S)=d\leq 1$ for all $T,S$. This can easily be seen by restricting those modules to the $d$-Kronecker subquiver this gives. For species there should be a similar criterion, but instead of $\operatorname{dim}_k$ one should use $\operatorname{dim}_{\operatorname{End}(S)}$ and $\operatorname{dim}_{\operatorname{End}(T)}$. $\endgroup$
    – Julian Kuelshammer
    Commented Dec 3, 2018 at 10:38

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