Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 2 / 4 / ii / Solution

Take the Kronecker quiver, with two arrows $\alpha ,\beta :1\to 2$. There are no paths of length two, so its path algebra has basis $e_1,e_2,\alpha ,\beta$ and dimension four.

For each $\lambda \in k$, take a quiver representation with one-dimensional spaces at both vertices and arrow maps $(1,\lambda )$. An endomorphism consists of scalars $a,b$ with $b=a$ from the first arrow; the second imposes no further condition. Its endomorphism ring is therefore $k$, which has no nontrivial idempotent. Thus the representation is indecomposable. An isomorphism to the representation $(1,\mu )$ again requires $b=a\ne 0$, and the second arrow then gives $\lambda =\mu$. Because an algebraically closed field is infinite, this gives infinitely many pairwise nonisomorphic indecomposables. Hence