Derived tubular algebras and APR-tilts (Q2717689)

Let \(A\) be a finite-dimensional algebra over an algebraically closed field and suppose there exists a simple projective \(A\)-module \(S\). Let \(\widehat P\) be the sum of one copy of each indecomposable projective module not isomorphic to \(S\). The module \(T=\tau_A^{-1}S\oplus\widehat P\), where \(\tau_A\) denotes the Auslander-Reiten translation, is called an APR-tilting module. Dually, by using a simple injective module, one can define an APR-cotilting module. There are nice categorical relations between the algebras \(A\) and \(B=\text{End }T\), when \(T\) is an APR-(co)tilting module and this enables one to transfer informations from \(\text{mod }A\) to \(\text{mod }B\). So, it is interesting to know, for a well-known class of algebras, which algebras can be reached by an iteration of the above process of taking endomorphism rings of APR-(co)tilting modules. The main result here is that each derived tubular algebra can be transformed by a finite sequence of APR-tilts or APR-cotilts to a canonical algebra. This is no longer true if one starts with an algebra which is derived equivalent to a canonical algebra of domestic type.