I am reading Green, Hille & Schroll's paper Algebras and Varieties and find myself a bit confused with Proposition 5.1, which says:
Consider a field $K$, a quiver $Q$ and a tip-reduced nonempty set of paths $\mathcal{T}$ such that $kQ/\langle\mathcal{T}\rangle$ is finite dimensional. Then $\text{gldim}(KQ/\langle\mathcal{T}\rangle)=2$ if and only if there are no overlaps among elements of $\mathcal{T}$.
However, if $Q$ is the quiver with a vertex and $1$ loop $x$ and $\mathcal{T}=\{x^2\}$, then the above seems to fail. What is it that I don't understand?