Die Vektorraumkategorie zu einem unzerlegbaren projektiven Modul einer tubularen Algebra. (The vector space category to an indecomposable projective module of a tubular algebra) (Q750570)

The author continues his study of the vectorspace category (add \(S_ x^ A,Hom_ A(P(x)\),-)) associated to a point x of the quiver of a given algebra A. P(x) denotes the indecomposable left A-module induced by x and \(S_ x^ A\) is a set of representatives of the isomorphism classes of all indecomposable modules X in A-mod (the category of finite dimensional left A-modules) satisfying \(X\not\cong P(x)\) and \(Hom_ A(P(x),X)\neq 0=Hom_ A(P(x),\tau_ AX)\) \((\tau_ A\) is the Auslander- Reiten translation). He considers the case that A is a tubular algebra and x belongs simultaneously to \(A_ 0\) and \(A_{\infty}\) (for definitions see [\textit{C. M. Ringel}, Tame algebras and integral quadratic forms (Lect. Notes Math. 1099, 1984; Zbl 0546.16013)]). As first result an injection from the critical subcategories of (add \(S_ x^ A,Hom_ A(P(x)\),-)) to the index set \({\mathbb{Q}}_ 0^{\infty}\) of the tubular families of A is given and it is discussed how the direct sum of the modules in \(S_ x^ A\) can be completed to a tilting module. Then it is shown that the union of all \(X\in S_ x^ A\) occurring in some critical subcategory of (add \(S_ x^ A,Hom_ A(P(x)\),-)) form a convex subcategory of \(S_ x^ A\) with respect to the partial order induced by \(X\leq Y\) if \(Hom_ A(P(x),Hom_ A(X,Y))\neq 0\). Finally it is proved that every indecomposable A-module satisfying \(Hom_ A(P(x),X)\neq 0\) which does not belong to the nonhomogeneous tubes of the last tubular family \({\mathcal T}_{\infty}\) of A lies in the image of the fiber sum functor with respect to P(x) (see [\textit{P. Dräxler}, J. Algebra 113, 430-437 (1988; Zbl 0659.16020)]). The presented results generalize in a natural way previous results of the author about tame concealed algebras which will appear in J. Algebra.