Kronecker quiver and tensor product

Question

Good day, based on representation theory of Assem,

and the definition of tensor product

I need to find $\varphi^1_{21}$ and its domain in the kronecker quiver. I did the next:

but Im confused to find $F_0$ and therefore $F/F_0$ to deduce the domain $\varepsilon _2KQ\varepsilon _1\otimes \varepsilon _1KQ\varepsilon _1$. Depends it of the field $K$ and of the external operation $ka$, $k\in K, a\in \varepsilon _2KQ\varepsilon _1$?

Answer 1

As $\varepsilon_{2}KQ \varepsilon_1 \cong K^2$, $\varepsilon_{i}KQ \varepsilon_i \cong K$, then $\varphi_{21}^1:K^2 \otimes K\rightarrow K^2$ and for properties of tensor product $K^2 \otimes K\cong K^2.$