The vector space category for an indecomposable projective module of an algebra (Q1176669)

Let \(\kappa\) be a field and \(A\) a finite dimensional \(\kappa\)-algebra; with an indecomposable projective \(A\)-module \(P(x)\), \textit{C. M. Ringel} and \textit{D. Vossieck} [Proc. Lond. Math. Soc., III. Ser. 54, 216-246 (1987; Zbl 0621.16031)] have associated a vector space category \({\mathcal S}_ x^ A=(\text{add}{\mathcal S}_ x^ A,\text{Hom}_ A(P(x),-1)\) with \({\mathcal S}^ A_ x\) a set of representatives of isomorphism classes of indecomposable \(A\)-modules \(M\) with \(\text{Hom}_ A(P(x),M)\neq 0\neq\text{Hom}_ A(P(x),\tau M)\), where \(\tau\) is the Auslander-Reiten translate, and studied it in case \(A\) is representation directed. The author studies critical vector space categories of \({\mathcal S}^ A_ x\) in general and pays special attention to the case where \(A\) is tame concealed. In this latter case \(\{M\in\bmod A:\text{Hom}_ A(P(x),M)\neq 0\}\) is a fibre sum over \(P(x)\) in the sense of \textit{P. Dräxler} [J. Algebra 113, 430-437 (1988; Zbl 0659.16020)].