Skip to main content
edited tags
Link
It'sMe
  • 839
  • 1
  • 8
  • 16
added 4 characters in body
Source Link
It'sMe
  • 839
  • 1
  • 8
  • 16

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:

  1. $B$ is a brick, i.e., $\operatorname{End}_{A}(B)\cong\mathbb{K}$.
  2. Projective dimension of $B$ is at most $1$ and injective dimension of $B$ is at most 1.
  3. $\operatorname{Ext}^{i}(B,B)=0$, for all $i\geq1$.
  4. $B$ is the labelling of the $(n-1)-$dimensional wall, saycall 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:enter image description here

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 $T(B)\cap\,^\bot F(B)$ enter image description here

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:

  1. $B$ is a brick, i.e., $\operatorname{End}_{A}(B)\cong\mathbb{K}$.
  2. Projective dimension of $B$ is at most $1$ and injective dimension of $B$ is at most 1.
  3. $\operatorname{Ext}^{i}(B,B)=0$, for all $i\geq1$.
  4. $B$ is the labelling of the $(n-1)-$dimensional wall, say $\Theta_{B}$, 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:enter image description here

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 $T(B)\cap\,^\bot F(B)$ enter image description here

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:

  1. $B$ is a brick, i.e., $\operatorname{End}_{A}(B)\cong\mathbb{K}$.
  2. Projective dimension of $B$ is at most $1$ and injective dimension of $B$ is at most 1.
  3. $\operatorname{Ext}^{i}(B,B)=0$, for all $i\geq1$.
  4. $B$ 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:enter image description here

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 $T(B)\cap\,^\bot F(B)$ enter image description here

Source Link
It'sMe
  • 839
  • 1
  • 8
  • 16

Prove that $B$ is a directing module

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:

  1. $B$ is a brick, i.e., $\operatorname{End}_{A}(B)\cong\mathbb{K}$.
  2. Projective dimension of $B$ is at most $1$ and injective dimension of $B$ is at most 1.
  3. $\operatorname{Ext}^{i}(B,B)=0$, for all $i\geq1$.
  4. $B$ is the labelling of the $(n-1)-$dimensional wall, say $\Theta_{B}$, 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:enter image description here

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 $T(B)\cap\,^\bot F(B)$ enter image description here