Homogeneous tubes over one-point extensions (Q1821855)

Let R be a finite dimensional algebra of the form \(R=\left( \begin{matrix} k\quad M\\ 0\quad A\end{matrix} \right)\), where k is an algebraically closed field, A is an algebra and \({}_ kM_ A\) is a nonzero k-A- bimodule. A description of the Auslander-Reiten translate \(\tau (X_ R)\) of a right nonprojective indecomposable R-module \(X_ R\) is given. Necessary and sufficient conditions for an indecomposable nonprojective R-module \(X_ R=(k,X'_ A\), \(\phi\) : \(k\otimes_ kM_ A\to X'_ A)\) to have the property \(\tau (X_ R)\cong X_ R\) are also presented. It is observed that any such module \(X_ R\) is quasi-simple and a description of the Auslander-Reiten component containing \(X_ R\) is given.