On wild tilted algebras which are squids
The author calls a finite-dimensional algebra B over an algebraically closed field an r-squid, if B is a one-point extension \(B=D[R]\), where D is a hereditary algebra with exactly one simple injective module S, and R is an indecomposable D-module with top R\(=S^ r\). She proves as main result: Let C be a wild hereditary algebra with at least three simple modules. For all \(N\in {\mathbb{N}}\) there is a tilted algebra B of C which is an r-squid for some \(r\geq N\). Moreover B has exactly one preprojective component, namely the preprojective component of D and exactly one preinjective component, which is the preinjective component of the generalized Kronecker-algebra \(K_ r.\) The idea of the proof is the following: One may assume that the algebra C is a one-point co-extension of a representation-infinite algebra A. Choosing a suitable indecomposable module U in A-mod with \(Ext^ 1(U,U)=0\), consider the minimal projective generator M of U in A-mod. Then there exists exactly one indecomposable C-module N such that \(T=U\oplus M\oplus N\) is a tilting module in C-mod and \(B=End(T)\) has the desired properties. A consequence of this concept is the result that almost all preprojective A-modules are quasi-simple regular C-modules. In the short first part the author proves some interesting lemmas, which seem to be useful for many purposes.
scientific article; zbMATH DE number 4200411
scientific article