Let $A\cong\mathbb{K}Q/I$ be a finite dimensional, associative, basic $\mathbb{K}$ algebra, where $\mathbb{K}$ is algebraically closed and $Q$ is a finite Gabriel quiver on $n$ vertices and $I\subseteq\mathbb{K}Q$ is an admissible ideal. Assume that $A$ is a tame algebra and $Q$ is acyclic. Let $B$ be an $A-$module such that $B$ has following properties:
- $B$ is a brick, i.e., $\operatorname{End}_{A}(B)\cong\mathbb{K}$.
- Projective dimension of $B$ is at most $1$ and injective dimension of $B$ is at most 1.
- $\operatorname{Ext}^{i}(B,B)=0$, for all $i\geq1$.
- $B$ is the labelling of the $(n-1)-$dimensional wall, call it $\Theta_{B}$, in the wall and chamber structure of $A$ such that the wall $\Theta_{B}$ is adjacent to two or more chambers.
Prove that $B$ is a directing module.
Here's the definition of directing modules:
I've been trying to use the following lemma from the book "tame algebras and integral quadratic forms" by C. M. Ringel to prove it, but can't figure out what $X$ and $Y$ should be. My guess for $N^{0}$ is the torsion class $T(B)\cap\,^\bot F(B)$
